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Theorem hashf1lem2 14179

Description: Lemma for hashf1 14180. (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 3944	. 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 9124	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝐵 ↑_m 𝐴) ∈ Fin)
5	2, 3, 4	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐵 ↑_m 𝐴) ∈ Fin)
6		f1f 6679	. . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
7	2, 3	elmapd 8638	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑_m 𝐴) ↔ 𝑓:𝐴⟶𝐵))
8	6, 7	syl5ibr 245	. . . . 5 ⊢ (𝜑 → (𝑓:𝐴–1-1→𝐵 → 𝑓 ∈ (𝐵 ↑_m 𝐴)))
9	8	abssdv 4003	. . . 4 ⊢ (𝜑 → {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ (𝐵 ↑_m 𝐴))
10	5, 9	ssfid 9051	. . 3 ⊢ (𝜑 → {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ∈ Fin)
11		sseq1 3947	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ ∅ ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))
12		eleq2 2828	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ ∅))
13		noel 4265	. . . . . . . . . . . . . 14 ⊢ ¬ (𝑓 ↾ 𝐴) ∈ ∅
14	13	pm2.21i 119	. . . . . . . . . . . . 13 ⊢ ((𝑓 ↾ 𝐴) ∈ ∅ → 𝑓 ∈ ∅)
15	12, 14	syl6bi 252	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ((𝑓 ↾ 𝐴) ∈ 𝑥 → 𝑓 ∈ ∅))
16	15	adantrd 492	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓 ∈ ∅))
17	16	abssdv 4003	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ ∅)
18		ss0 4333	. . . . . . . . . 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 6787	. . . . . . . 8 ⊢ (𝑥 = ∅ → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (♯‘∅))
21		hash0 14091	. . . . . . . 8 ⊢ (♯‘∅) = 0
22	20, 21	eqtrdi 2795	. . . . . . 7 ⊢ (𝑥 = ∅ → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = 0)
23		fveq2 6783	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
24	23, 21	eqtrdi 2795	. . . . . . . 8 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
25	24	oveq2d 7300	. . . . . . 7 ⊢ (𝑥 = ∅ → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) = (((♯‘𝐵) − (♯‘𝐴)) · 0))
26	22, 25	eqeq12d 2755	. . . . . 6 ⊢ (𝑥 = ∅ → ((♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) ↔ 0 = (((♯‘𝐵) − (♯‘𝐴)) · 0)))
27	11, 26	imbi12d 345	. . . . 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 341	. . . 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 3947	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ 𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))
30		eleq2 2828	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ 𝑦))
31	30	anbi1d 630	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)))
32	31	abbidv 2808	. . . . . . . 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 6787	. . . . . . 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 6783	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
35	34	oveq2d 7300	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑦)))
36	33, 35	eqeq12d 2755	. . . . . 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 345	. . . . 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 341	. . . 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 3947	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))
40		eleq2 2828	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎})))
41	40	anbi1d 630	. . . . . . . . 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 2808	. . . . . . . 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 6787	. . . . . . 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 6783	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑎})))
45	44	oveq2d 7300	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘(𝑦 ∪ {𝑎}))))
46	43, 45	eqeq12d 2755	. . . . . 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 345	. . . . 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 341	. . . 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 3947	. . . . . 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 6674	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑦 → (𝑓:𝐴–1-1→𝐵 ↔ 𝑦:𝐴–1-1→𝐵))
51	50	cbvabv 2812	. . . . . . . . . 10 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵}
52	51	eqeq2i 2752	. . . . . . . . 9 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ 𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵})
53		ssun1 4107	. . . . . . . . . . . . . . 15 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑧})
54		f1ssres 6687	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ∧ 𝐴 ⊆ (𝐴 ∪ {𝑧})) → (𝑓 ↾ 𝐴):𝐴–1-1→𝐵)
55	53, 54	mpan2 688	. . . . . . . . . . . . . 14 ⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → (𝑓 ↾ 𝐴):𝐴–1-1→𝐵)
56		vex 3437	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
57	56	resex 5942	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ↾ 𝐴) ∈ V
58		f1eq1 6674	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑓 ↾ 𝐴) → (𝑦:𝐴–1-1→𝐵 ↔ (𝑓 ↾ 𝐴):𝐴–1-1→𝐵))
59	57, 58	elab 3610	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ↾ 𝐴) ∈ {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} ↔ (𝑓 ↾ 𝐴):𝐴–1-1→𝐵)
60	55, 59	sylibr 233	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → (𝑓 ↾ 𝐴) ∈ {𝑦 ∣ 𝑦:𝐴–1-1→𝐵})
61		eleq2 2828	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ {𝑦 ∣ 𝑦:𝐴–1-1→𝐵}))
62	60, 61	syl5ibr 245	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → (𝑓 ↾ 𝐴) ∈ 𝑥))
63	62	pm4.71rd 563	. . . . . . . . . . 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 222	. . . . . . . . . 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 2808	. . . . . . . . 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 216	. . . . . . . 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 6787	. . . . . . 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 6783	. . . . . . . 8 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (♯‘𝑥) = (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))
69	68	oveq2d 7300	. . . . . . 7 ⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (((♯‘𝐵) − (♯‘𝐴)) · (♯‘𝑥)) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵})))
70	67, 69	eqeq12d 2755	. . . . . 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 345	. . . . 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 341	. . . 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 14080	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
74	2, 73	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ₀)
75	74	nn0cnd 12304	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝐵) ∈ ℂ)
76		hashcl 14080	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
77	3, 76	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
78	77	nn0cnd 12304	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝐴) ∈ ℂ)
79	75, 78	subcld 11341	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℂ)
80	79	mul01d 11183	. . . . . 6 ⊢ (𝜑 → (((♯‘𝐵) − (♯‘𝐴)) · 0) = 0)
81	80	eqcomd 2745	. . . . 5 ⊢ (𝜑 → 0 = (((♯‘𝐵) − (♯‘𝐴)) · 0))
82	81	a1d 25	. . . 4 ⊢ (𝜑 → (∅ ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → 0 = (((♯‘𝐵) − (♯‘𝐴)) · 0)))
83		ssun1 4107	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑎})
84		sstr 3930	. . . . . . . . 9 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑎}) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → 𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
85	83, 84	mpan 687	. . . . . . . 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 7291	. . . . . . . . . 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 4084	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) ∈ {𝑎}))
89	57	elsn 4577	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ↾ 𝐴) ∈ {𝑎} ↔ (𝑓 ↾ 𝐴) = 𝑎)
90	89	orbi2i 910	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) ∈ {𝑎}) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) = 𝑎))
91	88, 90	bitri 274	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) = 𝑎))
92	91	anbi1i 624	. . . . . . . . . . . . . . . . 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 1006	. . . . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . . . 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 4233	. . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . 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 6786	. . . . . . . . . . . . 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 8843	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑧} ∈ Fin
100		unfi 8964	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
101	3, 99, 100	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
102		mapvalg 8634	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∪ {𝑧}) ∈ Fin) → (𝐵 ↑_m (𝐴 ∪ {𝑧})) = {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵})
103	2, 101, 102	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ↑_m (𝐴 ∪ {𝑧})) = {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵})
104		mapfi 9124	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∪ {𝑧}) ∈ Fin) → (𝐵 ↑_m (𝐴 ∪ {𝑧})) ∈ Fin)
105	2, 101, 104	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ↑_m (𝐴 ∪ {𝑧})) ∈ Fin)
106	103, 105	eqeltrrd 2841	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵} ∈ Fin)
107		f1f 6679	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
108	107	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
109	108	ss2abi 4001	. . . . . . . . . . . . . . . 16 ⊢ {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵}
110		ssfi 8965	. . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin)
112	111	adantr 481	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
114	113	ss2abi 4001	. . . . . . . . . . . . . . . 16 ⊢ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵}
115		ssfi 8965	. . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin)
117	116	adantr 481	. . . . . . . . . . . . . 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 4234	. . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ¬ 𝑎 ∈ 𝑦)
120		abn0 4315	. . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑓 ↾ 𝐴) = 𝑎)
122		simpll 764	. . . . . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑎 ∈ 𝑦)
124	123	exlimiv 1934	. . . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . . 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 2973	. . . . . . . . . . . . . . . 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	eqtrid 2791	. . . . . . . . . . . . . 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 14106	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 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 2791	. . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
133	132	unssbd 4123	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → {𝑎} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
134		vex 3437	. . . . . . . . . . . . . . . . 17 ⊢ 𝑎 ∈ V
135	134	snss 4720	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ {𝑎} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
136	133, 135	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → 𝑎 ∈ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})
137		f1eq1 6674	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑎 → (𝑓:𝐴–1-1→𝐵 ↔ 𝑎:𝐴–1-1→𝐵))
138	134, 137	elab 3610	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ 𝑎:𝐴–1-1→𝐵)
139	136, 138	sylib 217	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → 𝑎:𝐴–1-1→𝐵)
140	78	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘𝐴) ∈ ℂ)
141	116	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin)
142		hashcl 14080	. . . . . . . . . . . . . . . . . 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 12304	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) ∈ ℂ)
145	140, 144	pncan2d 11343	. . . . . . . . . . . . . . 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 6736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎:𝐴–1-1→𝐵 → 𝑎:𝐴–1-1-onto→ran 𝑎)
147	146	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝑎:𝐴–1-1-onto→ran 𝑎)
148		f1oen3g 8763	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 ∈ V ∧ 𝑎:𝐴–1-1-onto→ran 𝑎) → 𝐴 ≈ ran 𝑎)
149	134, 147, 148	sylancr 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝐴 ≈ ran 𝑎)
150		hasheni 14071	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝐴 ∈ Fin)
153	2	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝐵 ∈ Fin)
154		hashf1lem2.3	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
155	154	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ¬ 𝑧 ∈ 𝐴)
156		hashf1lem2.4	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))
157	156	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))
158		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝑎:𝐴–1-1→𝐵)
159	152, 153, 155, 157, 158	hashf1lem1 14177	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝑎))
160		hasheni 14071	. . . . . . . . . . . . . . . . . . 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 7302	. . . . . . . . . . . . . . . . 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 6679	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎:𝐴–1-1→𝐵 → 𝑎:𝐴⟶𝐵)
164	163	frnd 6617	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎:𝐴–1-1→𝐵 → ran 𝑎 ⊆ 𝐵)
165	164	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ran 𝑎 ⊆ 𝐵)
166	153, 165	ssfid 9051	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ran 𝑎 ∈ Fin)
167		diffi 8971	. . . . . . . . . . . . . . . . . . 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 4406	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran 𝑎 ∩ (𝐵 ∖ ran 𝑎)) = ∅
170	169	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (ran 𝑎 ∩ (𝐵 ∖ ran 𝑎)) = ∅)
171		hashun 14106	. . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘(ran 𝑎 ∪ (𝐵 ∖ ran 𝑎))) = ((♯‘ran 𝑎) + (♯‘(𝐵 ∖ ran 𝑎))))
173		undif 4416	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran 𝑎 ⊆ 𝐵 ↔ (ran 𝑎 ∪ (𝐵 ∖ ran 𝑎)) = 𝐵)
174	165, 173	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (ran 𝑎 ∪ (𝐵 ∖ ran 𝑎)) = 𝐵)
175	174	fveq2d 6787	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘(ran 𝑎 ∪ (𝐵 ∖ ran 𝑎))) = (♯‘𝐵))
176	162, 172, 175	3eqtr2d 2785	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ((♯‘𝐴) + (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) = (♯‘𝐵))
177	176	oveq1d 7299	. . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (♯‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ((♯‘𝐵) − (♯‘𝐴)))
179	139, 178	sylan2 593	. . . . . . . . . . . . 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 7300	. . . . . . . . . . . 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 2779	. . . . . . . . . . 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 14116	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ V → ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑎})) = ((♯‘𝑦) + 1)))
183	182	elv 3439	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑎})) = ((♯‘𝑦) + 1))
184	183	ad2antrl 725	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (♯‘(𝑦 ∪ {𝑎})) = ((♯‘𝑦) + 1))
185	184	oveq2d 7300	. . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℂ)
187		simprll 776	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → 𝑦 ∈ Fin)
188		hashcl 14080	. . . . . . . . . . . . . . 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 12304	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (♯‘𝑦) ∈ ℂ)
191		1cnd 10979	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → 1 ∈ ℂ)
192	186, 190, 191	adddid 11008	. . . . . . . . . . . 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 11001	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (((♯‘𝐵) − (♯‘𝐴)) · 1) = ((♯‘𝐵) − (♯‘𝐴)))
194	193	oveq2d 7300	. . . . . . . . . . . 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 2783	. . . . . . . . . . 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 2755	. . . . . . . . . 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 245	. . . . . . . . 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 457	. . . . . . . 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 414	. . . . 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 8957	. . 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 396 ∨ wo 844 = wceq 1539 ∃wex 1782 ∈ wcel 2107 {cab 2716 ≠ wne 2944 Vcvv 3433 ∖ cdif 3885 ∪ cun 3886 ∩ cin 3887 ⊆ wss 3888 ∅c0 4257 {csn 4562 class class class wbr 5075 ran crn 5591 ↾ cres 5592 ⟶wf 6433 –1-1→wf1 6434 –1-1-onto→wf1o 6436 ‘cfv 6437 (class class class)co 7284 ↑_m cmap 8624 ≈ cen 8739 Fincfn 8742 ℂcc 10878 0cc0 10880 1c1 10881 + caddc 10883 · cmul 10885 ≤ cle 11019 − cmin 11214 ℕ₀cn0 12242 ♯chash 14053

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-rep 5210 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-om 7722 df-1st 7840 df-2nd 7841 df-frecs 8106 df-wrecs 8137 df-recs 8211 df-rdg 8250 df-1o 8306 df-oadd 8310 df-er 8507 df-map 8626 df-pm 8627 df-en 8743 df-dom 8744 df-sdom 8745 df-fin 8746 df-dju 9668 df-card 9706 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-nn 11983 df-n0 12243 df-z 12329 df-uz 12592 df-fz 13249 df-hash 14054

This theorem is referenced by: hashf1 14180

W3C validator