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Theorem hashf1lem2 13813

Description: Lemma for hashf1 13814. (Contributed by Mario Carneiro, 17-Apr-2015.)

**Hypotheses**
Ref	Expression
hashf1lem2.1	⊢ (𝜑 → 𝐴 ∈ Fin)
hashf1lem2.2	⊢ (𝜑 → 𝐵 ∈ Fin)
hashf1lem2.3	⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
hashf1lem2.4	⊢ (𝜑 → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))

**Assertion**
Ref	Expression
hashf1lem2	⊢ (𝜑 → (♯‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵})))

Distinct variable groups: 𝑧,𝑓 𝐴,𝑓 𝐵,𝑓 𝜑,𝑓 Allowed substitution hints: 𝜑(𝑧) 𝐴(𝑧) 𝐵(𝑧)

Proof of Theorem hashf1lem2 Dummy variables 𝑎 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3988	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}
2		hashf1lem2.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		hashf1lem2.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
4		mapfi 8819	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝐵 ↑_m 𝐴) ∈ Fin)
5	2, 3, 4	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐵 ↑_m 𝐴) ∈ Fin)
6		f1f 6574	. . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
7	2, 3	elmapd 8419	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑_m 𝐴) ↔ 𝑓:𝐴⟶𝐵))
8	6, 7	syl5ibr 248	. . . . 5 ⊢ (𝜑 → (𝑓:𝐴–1-1→𝐵 → 𝑓 ∈ (𝐵 ↑_m 𝐴)))
9	8	abssdv 4044	. . . 4 ⊢ (𝜑 → {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ (𝐵 ↑_m 𝐴))
10	5, 9	ssfid 8740	. . 3 ⊢ (𝜑 → {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ∈ Fin)
11		sseq1 3991	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ ∅ ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))
12		eleq2 2901	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ ∅))
13		noel 4295	. . . . . . . . . . . . . 14 ⊢ ¬ (𝑓 ↾ 𝐴) ∈ ∅
14	13	pm2.21i 119	. . . . . . . . . . . . 13 ⊢ ((𝑓 ↾ 𝐴) ∈ ∅ → 𝑓 ∈ ∅)
15	12, 14	syl6bi 255	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ((𝑓 ↾ 𝐴) ∈ 𝑥 → 𝑓 ∈ ∅))
16	15	adantrd 494	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓 ∈ ∅))
17	16	abssdv 4044	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ ∅)
18		ss0 4351	. . . . . . . . . 10 ⊢ ({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ ∅ → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = ∅)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ (𝑥 = ∅ → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = ∅)
20	19	fveq2d 6673	. . . . . . . 8 ⊢ (𝑥 = ∅ → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (♯‘∅))
21		hash0 13727	. . . . . . . 8 ⊢ (♯‘∅) = 0
22	20, 21	syl6eq 2872	. . . . . . 7 ⊢ (𝑥 = ∅ → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = 0)
23		fveq2 6669	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
24	23, 21	syl6eq 2872	. . . . . . . 8 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
25	24	oveq2d 7171	. . . . . . 7 ⊢ (𝑥 = ∅ → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) = (((♯‘𝐵) − (♯‘𝐴)) · 0))
26	22, 25	eqeq12d 2837	. . . . . 6 ⊢ (𝑥 = ∅ → ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) ↔ 0 = (((♯‘𝐵) − (♯‘𝐴)) · 0)))
27	11, 26	imbi12d 347	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥))) ↔ (∅ ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → 0 = (((♯‘𝐵) − (♯‘𝐴)) · 0))))
28	27	imbi2d 343	. . . 4 ⊢ (𝑥 = ∅ → ((𝜑 → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)))) ↔ (𝜑 → (∅ ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → 0 = (((♯‘𝐵) − (♯‘𝐴)) · 0)))))
29		sseq1 3991	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ 𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))
30		eleq2 2901	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ 𝑦))
31	30	anbi1d 631	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)))
32	31	abbidv 2885	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})
33	32	fveq2d 6673	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}))
34		fveq2 6669	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
35	34	oveq2d 7171	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)))
36	33, 35	eqeq12d 2837	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) ↔ (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦))))
37	29, 36	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥))) ↔ (𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)))))
38	37	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)))) ↔ (𝜑 → (𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦))))))
39		sseq1 3991	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))
40		eleq2 2901	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎})))
41	40	anbi1d 631	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)))
42	41	abbidv 2885	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})
43	42	fveq2d 6673	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}))
44		fveq2 6669	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑎})))
45	44	oveq2d 7171	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎}))))
46	43, 45	eqeq12d 2837	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) ↔ (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎})))))
47	39, 46	imbi12d 347	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → ((𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥))) ↔ ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎}))))))
48	47	imbi2d 343	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → ((𝜑 → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)))) ↔ (𝜑 → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎})))))))
49		sseq1 3991	. . . . . 6 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))
50		f1eq1 6569	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑦 → (𝑓:𝐴–1-1→𝐵 ↔ 𝑦:𝐴–1-1→𝐵))
51	50	cbvabv 2889	. . . . . . . . . 10 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵}
52	51	eqeq2i 2834	. . . . . . . . 9 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ 𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵})
53		ssun1 4147	. . . . . . . . . . . . . . 15 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑧})
54		f1ssres 6581	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ∧ 𝐴 ⊆ (𝐴 ∪ {𝑧})) → (𝑓 ↾ 𝐴):𝐴–1-1→𝐵)
55	53, 54	mpan2 689	. . . . . . . . . . . . . 14 ⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → (𝑓 ↾ 𝐴):𝐴–1-1→𝐵)
56		vex 3497	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
57	56	resex 5898	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ↾ 𝐴) ∈ V
58		f1eq1 6569	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑓 ↾ 𝐴) → (𝑦:𝐴–1-1→𝐵 ↔ (𝑓 ↾ 𝐴):𝐴–1-1→𝐵))
59	57, 58	elab 3666	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ↾ 𝐴) ∈ {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} ↔ (𝑓 ↾ 𝐴):𝐴–1-1→𝐵)
60	55, 59	sylibr 236	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → (𝑓 ↾ 𝐴) ∈ {𝑦 ∣ 𝑦:𝐴–1-1→𝐵})
61		eleq2 2901	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ {𝑦 ∣ 𝑦:𝐴–1-1→𝐵}))
62	60, 61	syl5ibr 248	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → (𝑓 ↾ 𝐴) ∈ 𝑥))
63	62	pm4.71rd 565	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ↔ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)))
64	63	bicomd 225	. . . . . . . . . 10 ⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → (((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))
65	64	abbidv 2885	. . . . . . . . 9 ⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵})
66	52, 65	sylbi 219	. . . . . . . 8 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵})
67	66	fveq2d 6673	. . . . . . 7 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (♯‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}))
68		fveq2 6669	. . . . . . . 8 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘𝑥) = (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))
69	68	oveq2d 7171	. . . . . . 7 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵})))
70	67, 69	eqeq12d 2837	. . . . . 6 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) ↔ (♯‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))))
71	49, 70	imbi12d 347	. . . . 5 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → ((𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥))) ↔ ({𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵})))))
72	71	imbi2d 343	. . . 4 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → ((𝜑 → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)))) ↔ (𝜑 → ({𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))))))
73		hashcl 13716	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
74	2, 73	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ₀)
75	74	nn0cnd 11956	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝐵) ∈ ℂ)
76		hashcl 13716	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
77	3, 76	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
78	77	nn0cnd 11956	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝐴) ∈ ℂ)
79	75, 78	subcld 10996	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℂ)
80	79	mul01d 10838	. . . . . 6 ⊢ (𝜑 → (((♯‘𝐵) − (♯‘𝐴)) · 0) = 0)
81	80	eqcomd 2827	. . . . 5 ⊢ (𝜑 → 0 = (((♯‘𝐵) − (♯‘𝐴)) · 0))
82	81	a1d 25	. . . 4 ⊢ (𝜑 → (∅ ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → 0 = (((♯‘𝐵) − (♯‘𝐴)) · 0)))
83		ssun1 4147	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑎})
84		sstr 3974	. . . . . . . . 9 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑎}) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → 𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
85	83, 84	mpan 688	. . . . . . . 8 ⊢ ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → 𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
86	85	imim1i 63	. . . . . . 7 ⊢ ((𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦))) → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦))))
87		oveq1 7162	. . . . . . . . . 10 ⊢ ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)) → ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + ((♯‘𝐵) − (♯‘𝐴))) = ((((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)) + ((♯‘𝐵) − (♯‘𝐴))))
88		elun 4124	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) ∈ {𝑎}))
89	57	elsn 4581	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ↾ 𝐴) ∈ {𝑎} ↔ (𝑓 ↾ 𝐴) = 𝑎)
90	89	orbi2i 909	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) ∈ {𝑎}) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) = 𝑎))
91	88, 90	bitri 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) = 𝑎))
92	91	anbi1i 625	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) = 𝑎) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))
93		andir 1005	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) = 𝑎) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∨ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)))
94	92, 93	bitri 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∨ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)))
95	94	abbii 2886	. . . . . . . . . . . . . . 15 ⊢ {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = {𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∨ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))}
96		unab 4269	. . . . . . . . . . . . . . 15 ⊢ ({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∪ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = {𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∨ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))}
97	95, 96	eqtr4i 2847	. . . . . . . . . . . . . 14 ⊢ {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = ({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∪ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})
98	97	fveq2i 6672	. . . . . . . . . . . . 13 ⊢ (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (♯‘({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∪ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}))
99		snfi 8593	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑧} ∈ Fin
100		unfi 8784	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
101	3, 99, 100	sylancl 588	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
102		mapvalg 8415	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∪ {𝑧}) ∈ Fin) → (𝐵 ↑_m (𝐴 ∪ {𝑧})) = {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵})
103	2, 101, 102	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ↑_m (𝐴 ∪ {𝑧})) = {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵})
104		mapfi 8819	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∪ {𝑧}) ∈ Fin) → (𝐵 ↑_m (𝐴 ∪ {𝑧})) ∈ Fin)
105	2, 101, 104	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ↑_m (𝐴 ∪ {𝑧})) ∈ Fin)
106	103, 105	eqeltrrd 2914	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵} ∈ Fin)
107		f1f 6574	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
108	107	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
109	108	ss2abi 4042	. . . . . . . . . . . . . . . 16 ⊢ {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵}
110		ssfi 8737	. . . . . . . . . . . . . . . 16 ⊢ (({𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵} ∈ Fin ∧ {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵}) → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin)
111	106, 109, 110	sylancl 588	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin)
112	111	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin)
113	107	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
114	113	ss2abi 4042	. . . . . . . . . . . . . . . 16 ⊢ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵}
115		ssfi 8737	. . . . . . . . . . . . . . . 16 ⊢ (({𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵} ∈ Fin ∧ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵}) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin)
116	106, 114, 115	sylancl 588	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin)
117	116	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin)
118		inab 4270	. . . . . . . . . . . . . . 15 ⊢ ({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∩ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = {𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))}
119		simprlr 778	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ¬ 𝑎 ∈ 𝑦)
120		abn0 4335	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))} ≠ ∅ ↔ ∃𝑓(((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)))
121		simprl 769	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑓 ↾ 𝐴) = 𝑎)
122		simpll 765	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑓 ↾ 𝐴) ∈ 𝑦)
123	121, 122	eqeltrrd 2914	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑎 ∈ 𝑦)
124	123	exlimiv 1927	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑓(((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑎 ∈ 𝑦)
125	120, 124	sylbi 219	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))} ≠ ∅ → 𝑎 ∈ 𝑦)
126	125	necon1bi 3044	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑎 ∈ 𝑦 → {𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))} = ∅)
127	119, 126	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → {𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))} = ∅)
128	118, 127	syl5eq 2868	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∩ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ∅)
129		hashun 13742	. . . . . . . . . . . . . 14 ⊢ (({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin ∧ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin ∧ ({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∩ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ∅) → (♯‘({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∪ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) = ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})))
130	112, 117, 128, 129	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (♯‘({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∪ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) = ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})))
131	98, 130	syl5eq 2868	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})))
132		simpr 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
133	132	unssbd 4163	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → {𝑎} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
134		vex 3497	. . . . . . . . . . . . . . . . 17 ⊢ 𝑎 ∈ V
135	134	snss 4717	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ {𝑎} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
136	133, 135	sylibr 236	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → 𝑎 ∈ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
137		f1eq1 6569	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑎 → (𝑓:𝐴–1-1→𝐵 ↔ 𝑎:𝐴–1-1→𝐵))
138	134, 137	elab 3666	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ 𝑎:𝐴–1-1→𝐵)
139	136, 138	sylib 220	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → 𝑎:𝐴–1-1→𝐵)
140	78	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘𝐴) ∈ ℂ)
141	116	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin)
142		hashcl 13716	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) ∈ ℕ₀)
143	141, 142	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) ∈ ℕ₀)
144	143	nn0cnd 11956	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) ∈ ℂ)
145	140, 144	pncan2d 10998	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (((♯‘𝐴) + (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) − (♯‘𝐴)) = (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}))
146		f1f1orn 6625	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎:𝐴–1-1→𝐵 → 𝑎:𝐴–1-1-onto→ran 𝑎)
147	146	adantl 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝑎:𝐴–1-1-onto→ran 𝑎)
148		f1oen3g 8524	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 ∈ V ∧ 𝑎:𝐴–1-1-onto→ran 𝑎) → 𝐴 ≈ ran 𝑎)
149	134, 147, 148	sylancr 589	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝐴 ≈ ran 𝑎)
150		hasheni 13707	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ≈ ran 𝑎 → (♯‘𝐴) = (♯‘ran 𝑎))
151	149, 150	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘𝐴) = (♯‘ran 𝑎))
152	3	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝐴 ∈ Fin)
153	2	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝐵 ∈ Fin)
154		hashf1lem2.3	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
155	154	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ¬ 𝑧 ∈ 𝐴)
156		hashf1lem2.4	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))
157	156	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))
158		simpr 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝑎:𝐴–1-1→𝐵)
159	152, 153, 155, 157, 158	hashf1lem1 13812	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝑎))
160		hasheni 13707	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝑎) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (♯‘(𝐵 ∖ ran 𝑎)))
161	159, 160	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (♯‘(𝐵 ∖ ran 𝑎)))
162	151, 161	oveq12d 7173	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ((♯‘𝐴) + (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) = ((♯‘ran 𝑎) + (♯‘(𝐵 ∖ ran 𝑎))))
163		f1f 6574	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎:𝐴–1-1→𝐵 → 𝑎:𝐴⟶𝐵)
164	163	frnd 6520	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎:𝐴–1-1→𝐵 → ran 𝑎 ⊆ 𝐵)
165	164	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ran 𝑎 ⊆ 𝐵)
166	153, 165	ssfid 8740	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ran 𝑎 ∈ Fin)
167		diffi 8749	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Fin → (𝐵 ∖ ran 𝑎) ∈ Fin)
168	153, 167	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (𝐵 ∖ ran 𝑎) ∈ Fin)
169		disjdif 4420	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran 𝑎 ∩ (𝐵 ∖ ran 𝑎)) = ∅
170	169	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (ran 𝑎 ∩ (𝐵 ∖ ran 𝑎)) = ∅)
171		hashun 13742	. . . . . . . . . . . . . . . . . 18 ⊢ ((ran 𝑎 ∈ Fin ∧ (𝐵 ∖ ran 𝑎) ∈ Fin ∧ (ran 𝑎 ∩ (𝐵 ∖ ran 𝑎)) = ∅) → (♯‘(ran 𝑎 ∪ (𝐵 ∖ ran 𝑎))) = ((♯‘ran 𝑎) + (♯‘(𝐵 ∖ ran 𝑎))))
172	166, 168, 170, 171	syl3anc 1367	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘(ran 𝑎 ∪ (𝐵 ∖ ran 𝑎))) = ((♯‘ran 𝑎) + (♯‘(𝐵 ∖ ran 𝑎))))
173		undif 4429	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran 𝑎 ⊆ 𝐵 ↔ (ran 𝑎 ∪ (𝐵 ∖ ran 𝑎)) = 𝐵)
174	165, 173	sylib 220	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (ran 𝑎 ∪ (𝐵 ∖ ran 𝑎)) = 𝐵)
175	174	fveq2d 6673	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘(ran 𝑎 ∪ (𝐵 ∖ ran 𝑎))) = (♯‘𝐵))
176	162, 172, 175	3eqtr2d 2862	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ((♯‘𝐴) + (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) = (♯‘𝐵))
177	176	oveq1d 7170	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (((♯‘𝐴) + (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) − (♯‘𝐴)) = ((♯‘𝐵) − (♯‘𝐴)))
178	145, 177	eqtr3d 2858	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ((♯‘𝐵) − (♯‘𝐴)))
179	139, 178	sylan2 594	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ((♯‘𝐵) − (♯‘𝐴)))
180	179	oveq2d 7171	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) = ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + ((♯‘𝐵) − (♯‘𝐴))))
181	131, 180	eqtrd 2856	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + ((♯‘𝐵) − (♯‘𝐴))))
182		hashunsng 13752	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ V → ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑎})) = ((♯‘𝑦) + 1)))
183	182	elv 3499	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑎})) = ((♯‘𝑦) + 1))
184	183	ad2antrl 726	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (♯‘(𝑦 ∪ {𝑎})) = ((♯‘𝑦) + 1))
185	184	oveq2d 7171	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎}))) = (((♯‘𝐵) − (♯‘𝐴)) · ((♯‘𝑦) + 1)))
186	79	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℂ)
187		simprll 777	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → 𝑦 ∈ Fin)
188		hashcl 13716	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
189	187, 188	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (♯‘𝑦) ∈ ℕ₀)
190	189	nn0cnd 11956	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (♯‘𝑦) ∈ ℂ)
191		1cnd 10635	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → 1 ∈ ℂ)
192	186, 190, 191	adddid 10664	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (((♯‘𝐵) − (♯‘𝐴)) · ((♯‘𝑦) + 1)) = ((((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)) + (((♯‘𝐵) − (♯‘𝐴)) · 1)))
193	186	mulid1d 10657	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (((♯‘𝐵) − (♯‘𝐴)) · 1) = ((♯‘𝐵) − (♯‘𝐴)))
194	193	oveq2d 7171	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ((((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)) + (((♯‘𝐵) − (♯‘𝐴)) · 1)) = ((((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)) + ((♯‘𝐵) − (♯‘𝐴))))
195	185, 192, 194	3eqtrd 2860	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎}))) = ((((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)) + ((♯‘𝐵) − (♯‘𝐴))))
196	181, 195	eqeq12d 2837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎}))) ↔ ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + ((♯‘𝐵) − (♯‘𝐴))) = ((((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)) + ((♯‘𝐵) − (♯‘𝐴)))))
197	87, 196	syl5ibr 248	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎})))))
198	197	expr 459	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦)) → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎}))))))
199	198	a2d 29	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦)) → (((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦))) → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎}))))))
200	86, 199	syl5 34	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦)) → ((𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦))) → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎}))))))
201	200	expcom 416	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) → (𝜑 → ((𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦))) → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎})))))))
202	201	a2d 29	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) → ((𝜑 → (𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)))) → (𝜑 → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎})))))))
203	28, 38, 48, 72, 82, 202	findcard2s 8758	. . 3 ⊢ ({𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ∈ Fin → (𝜑 → ({𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵})))))
204	10, 203	mpcom 38	. 2 ⊢ (𝜑 → ({𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))))
205	1, 204	mpi 20	1 ⊢ (𝜑 → (♯‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 ≠ wne 3016 Vcvv 3494 ∖ cdif 3932 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 {csn 4566 class class class wbr 5065 ran crn 5555 ↾ cres 5556 ⟶wf 6350 –1-1→wf1 6351 –1-1-onto→wf1o 6353 ‘cfv 6354 (class class class)co 7155 ↑_m cmap 8405 ≈ cen 8505 Fincfn 8508 ℂcc 10534 0cc0 10536 1c1 10537 + caddc 10539 · cmul 10541 ≤ cle 10675 − cmin 10869 ℕ₀cn0 11896 ♯chash 13689

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-hash 13690

This theorem is referenced by: hashf1 13814

W3C validator