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Theorem hashf1lem2 14495

Description: Lemma for hashf1 14496. (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 3960	. 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 9306	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝐵 ↑_m 𝐴) ∈ Fin)
5	2, 3, 4	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐵 ↑_m 𝐴) ∈ Fin)
6		f1f 6776	. . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
7	2, 3	elmapd 8838	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑_m 𝐴) ↔ 𝑓:𝐴⟶𝐵))
8	6, 7	imbitrrid 249	. . . . 5 ⊢ (𝜑 → (𝑓:𝐴–1-1→𝐵 → 𝑓 ∈ (𝐵 ↑_m 𝐴)))
9	8	abssdv 4022	. . . 4 ⊢ (𝜑 → {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ (𝐵 ↑_m 𝐴))
10	5, 9	ssfid 9230	. . 3 ⊢ (𝜑 → {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ∈ Fin)
11		sseq1 3963	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ ∅ ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))
12		eleq2 2852	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ ∅))
13		noel 4292	. . . . . . . . . . . . . 14 ⊢ ¬ (𝑓 ↾ 𝐴) ∈ ∅
14	13	pm2.21i 120	. . . . . . . . . . . . 13 ⊢ ((𝑓 ↾ 𝐴) ∈ ∅ → 𝑓 ∈ ∅)
15	12, 14	biimtrdi 256	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ((𝑓 ↾ 𝐴) ∈ 𝑥 → 𝑓 ∈ ∅))
16	15	adantrd 496	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓 ∈ ∅))
17	16	abssdv 4022	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ ∅)
18		ss0 4360	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ (𝑥 = ∅ → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = ∅)
20	19	fveq2d 6887	. . . . . . . 8 ⊢ (𝑥 = ∅ → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (♯‘∅))
21		hash0 14405	. . . . . . . 8 ⊢ (♯‘∅) = 0
22	20, 21	eqtrdi 2814	. . . . . . 7 ⊢ (𝑥 = ∅ → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = 0)
23		fveq2 6883	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
24	23, 21	eqtrdi 2814	. . . . . . . 8 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
25	24	oveq2d 7428	. . . . . . 7 ⊢ (𝑥 = ∅ → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) = (((♯‘𝐵) − (♯‘𝐴)) · 0))
26	22, 25	eqeq12d 2779	. . . . . 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 3963	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ 𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))
30		eleq2 2852	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ 𝑦))
31	30	anbi1d 642	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)))
32	31	abbidv 2829	. . . . . . . 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 6887	. . . . . . 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 6883	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
35	34	oveq2d 7428	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)))
36	33, 35	eqeq12d 2779	. . . . . 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 3963	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))
40		eleq2 2852	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎})))
41	40	anbi1d 642	. . . . . . . . 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 2829	. . . . . . . 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 6887	. . . . . . 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 6883	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑎})))
45	44	oveq2d 7428	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎}))))
46	43, 45	eqeq12d 2779	. . . . . 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 3963	. . . . . 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 6771	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑦 → (𝑓:𝐴–1-1→𝐵 ↔ 𝑦:𝐴–1-1→𝐵))
51	50	cbvabv 2833	. . . . . . . . . 10 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵}
52	51	eqeq2i 2776	. . . . . . . . 9 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ 𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵})
53		ssun1 4132	. . . . . . . . . . . . . . 15 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑧})
54		f1ssres 6785	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ∧ 𝐴 ⊆ (𝐴 ∪ {𝑧})) → (𝑓 ↾ 𝐴):𝐴–1-1→𝐵)
55	53, 54	mpan2 703	. . . . . . . . . . . . . 14 ⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → (𝑓 ↾ 𝐴):𝐴–1-1→𝐵)
56		vex 3459	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
57	56	resex 6030	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ↾ 𝐴) ∈ V
58		f1eq1 6771	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑓 ↾ 𝐴) → (𝑦:𝐴–1-1→𝐵 ↔ (𝑓 ↾ 𝐴):𝐴–1-1→𝐵))
59	57, 58	elab 3639	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ↾ 𝐴) ∈ {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} ↔ (𝑓 ↾ 𝐴):𝐴–1-1→𝐵)
60	55, 59	sylibr 237	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → (𝑓 ↾ 𝐴) ∈ {𝑦 ∣ 𝑦:𝐴–1-1→𝐵})
61		eleq2 2852	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ {𝑦 ∣ 𝑦:𝐴–1-1→𝐵}))
62	60, 61	imbitrrid 249	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → (𝑓 ↾ 𝐴) ∈ 𝑥))
63	62	pm4.71rd 571	. . . . . . . . . . 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 226	. . . . . . . . . 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 2829	. . . . . . . . 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 220	. . . . . . . 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 6887	. . . . . . 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 6883	. . . . . . . 8 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘𝑥) = (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))
69	68	oveq2d 7428	. . . . . . 7 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵})))
70	67, 69	eqeq12d 2779	. . . . . 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 14394	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
74	2, 73	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ₀)
75	74	nn0cnd 12568	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝐵) ∈ ℂ)
76		hashcl 14394	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
77	3, 76	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
78	77	nn0cnd 12568	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝐴) ∈ ℂ)
79	75, 78	subcld 11570	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℂ)
80	79	mul01d 11410	. . . . . 6 ⊢ (𝜑 → (((♯‘𝐵) − (♯‘𝐴)) · 0) = 0)
81	80	eqcomd 2769	. . . . 5 ⊢ (𝜑 → 0 = (((♯‘𝐵) − (♯‘𝐴)) · 0))
82	81	a1d 26	. . . 4 ⊢ (𝜑 → (∅ ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → 0 = (((♯‘𝐵) − (♯‘𝐴)) · 0)))
83		ssun1 4132	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑎})
84		sstr 3946	. . . . . . . . 9 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑎}) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → 𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
85	83, 84	mpan 702	. . . . . . . 8 ⊢ ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → 𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
86	85	imim1i 64	. . . . . . 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 7419	. . . . . . . . . 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 4108	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) ∈ {𝑎}))
89	57	elsn 4605	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ↾ 𝐴) ∈ {𝑎} ↔ (𝑓 ↾ 𝐴) = 𝑎)
90	89	orbi2i 925	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) ∈ {𝑎}) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) = 𝑎))
91	88, 90	bitri 278	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) = 𝑎))
92	91	anbi1i 635	. . . . . . . . . . . . . . . . 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 1026	. . . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . 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 4262	. . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . 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 6886	. . . . . . . . . . . . 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 9041	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑧} ∈ Fin
100		unfi 9156	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
101	3, 99, 100	sylancl 597	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
102		mapvalg 8834	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∪ {𝑧}) ∈ Fin) → (𝐵 ↑_m (𝐴 ∪ {𝑧})) = {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵})
103	2, 101, 102	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ↑_m (𝐴 ∪ {𝑧})) = {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵})
104		mapfi 9306	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∪ {𝑧}) ∈ Fin) → (𝐵 ↑_m (𝐴 ∪ {𝑧})) ∈ Fin)
105	2, 101, 104	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ↑_m (𝐴 ∪ {𝑧})) ∈ Fin)
106	103, 105	eqeltrrd 2864	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵} ∈ Fin)
107		f1f 6776	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
108	107	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
109	108	ss2abi 4021	. . . . . . . . . . . . . . . 16 ⊢ {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵}
110		ssfi 9158	. . . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin)
112	111	adantr 485	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
114	113	ss2abi 4021	. . . . . . . . . . . . . . . 16 ⊢ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵}
115		ssfi 9158	. . . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin)
117	116	adantr 485	. . . . . . . . . . . . . 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 4263	. . . . . . . . . . . . . . 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 791	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ¬ 𝑎 ∈ 𝑦)
120		abn0 4342	. . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑓 ↾ 𝐴) = 𝑎)
122		simpll 778	. . . . . . . . . . . . . . . . . . . 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 2864	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑎 ∈ 𝑦)
124	123	exlimiv 1960	. . . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . 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 2986	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 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	eqtrid 2810	. . . . . . . . . . . . . 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 14420	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 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	eqtrid 2810	. . . . . . . . . . . 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 489	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
133	132	unssbd 4148	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → {𝑎} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
134		vex 3459	. . . . . . . . . . . . . . . . 17 ⊢ 𝑎 ∈ V
135	134	snss 4751	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ {𝑎} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
136	133, 135	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → 𝑎 ∈ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
137		f1eq1 6771	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑎 → (𝑓:𝐴–1-1→𝐵 ↔ 𝑎:𝐴–1-1→𝐵))
138	134, 137	elab 3639	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ 𝑎:𝐴–1-1→𝐵)
139	136, 138	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → 𝑎:𝐴–1-1→𝐵)
140	78	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘𝐴) ∈ ℂ)
141	116	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin)
142		hashcl 14394	. . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) ∈ ℕ₀)
144	143	nn0cnd 12568	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) ∈ ℂ)
145	140, 144	pncan2d 11572	. . . . . . . . . . . . . . 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 6834	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎:𝐴–1-1→𝐵 → 𝑎:𝐴–1-1-onto→ran 𝑎)
147	146	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝑎:𝐴–1-1-onto→ran 𝑎)
148		f1oen3g 8964	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 ∈ V ∧ 𝑎:𝐴–1-1-onto→ran 𝑎) → 𝐴 ≈ ran 𝑎)
149	134, 147, 148	sylancr 598	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝐴 ≈ ran 𝑎)
150		hasheni 14386	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ≈ ran 𝑎 → (♯‘𝐴) = (♯‘ran 𝑎))
151	149, 150	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘𝐴) = (♯‘ran 𝑎))
152	3	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝐴 ∈ Fin)
153	2	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝐵 ∈ Fin)
154		hashf1lem2.3	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
155	154	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ¬ 𝑧 ∈ 𝐴)
156		hashf1lem2.4	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))
157	156	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))
158		simpr 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝑎:𝐴–1-1→𝐵)
159	152, 153, 155, 157, 158	hashf1lem1 14494	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝑎))
160		hasheni 14386	. . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (♯‘(𝐵 ∖ ran 𝑎)))
162	151, 161	oveq12d 7430	. . . . . . . . . . . . . . . . 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 6776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎:𝐴–1-1→𝐵 → 𝑎:𝐴⟶𝐵)
164	163	frnd 6716	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎:𝐴–1-1→𝐵 → ran 𝑎 ⊆ 𝐵)
165	164	adantl 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ran 𝑎 ⊆ 𝐵)
166	153, 165	ssfid 9230	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ran 𝑎 ∈ Fin)
167		diffi 9160	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Fin → (𝐵 ∖ ran 𝑎) ∈ Fin)
168	153, 167	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (𝐵 ∖ ran 𝑎) ∈ Fin)
169		disjdif 4434	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran 𝑎 ∩ (𝐵 ∖ ran 𝑎)) = ∅
170	169	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (ran 𝑎 ∩ (𝐵 ∖ ran 𝑎)) = ∅)
171		hashun 14420	. . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘(ran 𝑎 ∪ (𝐵 ∖ ran 𝑎))) = ((♯‘ran 𝑎) + (♯‘(𝐵 ∖ ran 𝑎))))
173		undif 4444	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran 𝑎 ⊆ 𝐵 ↔ (ran 𝑎 ∪ (𝐵 ∖ ran 𝑎)) = 𝐵)
174	165, 173	sylib 221	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (ran 𝑎 ∪ (𝐵 ∖ ran 𝑎)) = 𝐵)
175	174	fveq2d 6887	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘(ran 𝑎 ∪ (𝐵 ∖ ran 𝑎))) = (♯‘𝐵))
176	162, 172, 175	3eqtr2d 2804	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ((♯‘𝐴) + (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) = (♯‘𝐵))
177	176	oveq1d 7427	. . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ((♯‘𝐵) − (♯‘𝐴)))
179	139, 178	sylan2 604	. . . . . . . . . . . . 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 7428	. . . . . . . . . . . 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 2798	. . . . . . . . . . 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 14430	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ V → ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑎})) = ((♯‘𝑦) + 1)))
183	182	elv 3460	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑎})) = ((♯‘𝑦) + 1))
184	183	ad2antrl 740	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (♯‘(𝑦 ∪ {𝑎})) = ((♯‘𝑦) + 1))
185	184	oveq2d 7428	. . . . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℂ)
187		simprll 790	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → 𝑦 ∈ Fin)
188		hashcl 14394	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
189	187, 188	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (♯‘𝑦) ∈ ℕ₀)
190	189	nn0cnd 12568	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (♯‘𝑦) ∈ ℂ)
191		1cnd 11203	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → 1 ∈ ℂ)
192	186, 190, 191	adddid 11234	. . . . . . . . . . . 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	mulridd 11227	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (((♯‘𝐵) − (♯‘𝐴)) · 1) = ((♯‘𝐵) − (♯‘𝐴)))
194	193	oveq2d 7428	. . . . . . . . . . . 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 2802	. . . . . . . . . . 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 2779	. . . . . . . . . 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	imbitrrid 249	. . . . . . . . 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 461	. . . . . . . 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 30	. . . . . . 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 35	. . . . . 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 418	. . . . 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 30	. . . 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 9151	. . 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 39	. 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 21	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 400 ∨ wo 860 = wceq 1570 ∃wex 1809 ∈ wcel 2143 {cab 2741 ≠ wne 2958 Vcvv 3455 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 {csn 4590 class class class wbr 5110 ran crn 5664 ↾ cres 5665 ⟶wf 6534 –1-1→wf1 6535 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7412 ↑_m cmap 8825 ≈ cen 8941 Fincfn 8944 ℂcc 11099 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 ≤ cle 11245 − cmin 11442 ℕ₀cn0 12505 ♯chash 14368

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-oadd 8458 df-er 8695 df-map 8827 df-pm 8828 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-dju 9888 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-n0 12506 df-z 12593 df-uz 12864 df-fz 13537 df-hash 14369

This theorem is referenced by: hashf1 14496

W3C validator