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Theorem hashbclem 14590

Description: Lemma for hashbc 14591: inductive step. (Contributed by Mario Carneiro, 13-Jul-2014.)

**Hypotheses**
Ref	Expression
hashbc.1	⊢ (𝜑 → 𝐴 ∈ Fin)
hashbc.2	⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
hashbc.3	⊢ (𝜑 → ∀𝑗 ∈ ℤ ((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}))
hashbc.4	⊢ (𝜑 → 𝐾 ∈ ℤ)

**Assertion**
Ref	Expression
hashbclem	⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}))

Distinct variable groups: 𝑥,𝑗,𝑧,𝐴 𝑗,𝐾,𝑥 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑧, 𝑗) 𝐾(𝑧)

Proof of Theorem hashbclem Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7426	. . . . . 6 ⊢ (𝑗 = 𝐾 → ((♯‘𝐴)C𝑗) = ((♯‘𝐴)C𝐾))
2		eqeq2 2773	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑥) = 𝐾))
3	2	rabbidv 3420	. . . . . . 7 ⊢ (𝑗 = 𝐾 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})
4	3	fveq2d 6887	. . . . . 6 ⊢ (𝑗 = 𝐾 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
5	1, 4	eqeq12d 2777	. . . . 5 ⊢ (𝑗 = 𝐾 → (((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})))
6		hashbc.3	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ ℤ ((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}))
7		hashbc.4	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
8	5, 6, 7	rspcdva 3578	. . . 4 ⊢ (𝜑 → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
9		ssun1 4124	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑧})
10	9	sspwi 4569	. . . . . . . . . . . 12 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ {𝑧})
11	10	sseli 3927	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}))
12	11	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}))
13		hashbc.2	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
14		elpwi 4564	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
15	14	ssneld 3933	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝐴 → (¬ 𝑧 ∈ 𝐴 → ¬ 𝑧 ∈ 𝑥))
16	13, 15	mpan9 516	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → ¬ 𝑧 ∈ 𝑥)
17	12, 16	jca 521	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥))
18		elpwi 4564	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑥 ⊆ (𝐴 ∪ {𝑧}))
19		uncom 4105	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∪ {𝑧}) = ({𝑧} ∪ 𝐴)
20	18, 19	sseqtrdi 3971	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑥 ⊆ ({𝑧} ∪ 𝐴))
21	20	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ⊆ ({𝑧} ∪ 𝐴))
22		disjsn 4672	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑥)
23	22	bilanri 512	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → (𝑥 ∩ {𝑧}) = ∅)
24		disjssun 4421	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ {𝑧}) = ∅ → (𝑥 ⊆ ({𝑧} ∪ 𝐴) ↔ 𝑥 ⊆ 𝐴))
25	23, 24	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → (𝑥 ⊆ ({𝑧} ∪ 𝐴) ↔ 𝑥 ⊆ 𝐴))
26	21, 25	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ⊆ 𝐴)
27		vex 3455	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
28	27	elpw 4561	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
29	26, 28	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ∈ 𝒫 𝐴)
30	29	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥)) → 𝑥 ∈ 𝒫 𝐴)
31	17, 30	impbida 813	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥)))
32	31	anbi1d 643	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (♯‘𝑥) = 𝐾) ↔ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾)))
33		anass 474	. . . . . . 7 ⊢ (((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾) ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)))
34	32, 33	bitrdi 290	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))))
35	34	rabbidva2 3415	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾} = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
36	35	fveq2d 6887	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
37	8, 36	eqtrd 2796	. . 3 ⊢ (𝜑 → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
38		oveq2 7426	. . . . . 6 ⊢ (𝑗 = (𝐾 − 1) → ((♯‘𝐴)C𝑗) = ((♯‘𝐴)C(𝐾 − 1)))
39		eqeq2 2773	. . . . . . . 8 ⊢ (𝑗 = (𝐾 − 1) → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑥) = (𝐾 − 1)))
40	39	rabbidv 3420	. . . . . . 7 ⊢ (𝑗 = (𝐾 − 1) → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})
41	40	fveq2d 6887	. . . . . 6 ⊢ (𝑗 = (𝐾 − 1) → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
42	38, 41	eqeq12d 2777	. . . . 5 ⊢ (𝑗 = (𝐾 − 1) → (((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})))
43		peano2zm 12732	. . . . . 6 ⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ)
44	7, 43	syl 18	. . . . 5 ⊢ (𝜑 → (𝐾 − 1) ∈ ℤ)
45	42, 6, 44	rspcdva 3578	. . . 4 ⊢ (𝜑 → ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
46		hashbc.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Fin)
47		pwfi 9303	. . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
48	46, 47	sylib 221	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝐴 ∈ Fin)
49		rabexg 5299	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ Fin → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ V)
50	48, 49	syl 18	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ V)
51		snfi 9064	. . . . . . . . 9 ⊢ {𝑧} ∈ Fin
52		unfi 9179	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
53	46, 51, 52	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
54		pwfi 9303	. . . . . . . 8 ⊢ ((𝐴 ∪ {𝑧}) ∈ Fin ↔ 𝒫 (𝐴 ∪ {𝑧}) ∈ Fin)
55	53, 54	sylib 221	. . . . . . 7 ⊢ (𝜑 → 𝒫 (𝐴 ∪ {𝑧}) ∈ Fin)
56		ssrab2 4028	. . . . . . 7 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})
57		ssfi 9181	. . . . . . 7 ⊢ ((𝒫 (𝐴 ∪ {𝑧}) ∈ Fin ∧ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})) → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
58	55, 56, 57	sylancl 598	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
59		fveqeq2 6892	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((♯‘𝑥) = (𝐾 − 1) ↔ (♯‘𝑢) = (𝐾 − 1)))
60	59	elrab 3645	. . . . . . 7 ⊢ (𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ↔ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)))
61		eleq2 2850	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ∪ {𝑧}) → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ (𝑢 ∪ {𝑧})))
62		fveqeq2 6892	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ∪ {𝑧}) → ((♯‘𝑥) = 𝐾 ↔ (♯‘(𝑢 ∪ {𝑧})) = 𝐾))
63	61, 62	anbi12d 644	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ∪ {𝑧}) → ((𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (♯‘(𝑢 ∪ {𝑧})) = 𝐾)))
64		elpwi 4564	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝒫 𝐴 → 𝑢 ⊆ 𝐴)
65	64	ad2antrl 741	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝑢 ⊆ 𝐴)
66		unss1 4131	. . . . . . . . . . 11 ⊢ (𝑢 ⊆ 𝐴 → (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
67	65, 66	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
68		vex 3455	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
69		vsnex 5393	. . . . . . . . . . . 12 ⊢ {𝑧} ∈ V
70	68, 69	unex 7759	. . . . . . . . . . 11 ⊢ (𝑢 ∪ {𝑧}) ∈ V
71	70	elpw 4561	. . . . . . . . . 10 ⊢ ((𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}) ↔ (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
72	67, 71	sylibr 237	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}))
73	46	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐴 ∈ Fin)
74	73, 65	ssfid 9253	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝑢 ∈ Fin)
75	51	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → {𝑧} ∈ Fin)
76	13	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ¬ 𝑧 ∈ 𝐴)
77	65, 76	ssneldd 3934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ¬ 𝑧 ∈ 𝑢)
78		disjsn 4672	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑢)
79	77, 78	sylibr 237	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∩ {𝑧}) = ∅)
80		hashun 14519	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝑢 ∩ {𝑧}) = ∅) → (♯‘(𝑢 ∪ {𝑧})) = ((♯‘𝑢) + (♯‘{𝑧})))
81	74, 75, 79, 80	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘(𝑢 ∪ {𝑧})) = ((♯‘𝑢) + (♯‘{𝑧})))
82		simprr 785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘𝑢) = (𝐾 − 1))
83		hashsng 14506	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → (♯‘{𝑧}) = 1)
84	83	elv 3456	. . . . . . . . . . . . 13 ⊢ (♯‘{𝑧}) = 1
85	84	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘{𝑧}) = 1)
86	82, 85	oveq12d 7436	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ((♯‘𝑢) + (♯‘{𝑧})) = ((𝐾 − 1) + 1))
87	7	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℤ)
88	87	zcnd 12797	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℂ)
89		ax-1cn 11251	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
90		npcan 11559	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 − 1) + 1) = 𝐾)
91	88, 89, 90	sylancl 598	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ((𝐾 − 1) + 1) = 𝐾)
92	81, 86, 91	3eqtrd 2800	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘(𝑢 ∪ {𝑧})) = 𝐾)
93		ssun2 4125	. . . . . . . . . . 11 ⊢ {𝑧} ⊆ (𝑢 ∪ {𝑧})
94		vex 3455	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
95	94	snss 4745	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑢 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝑢 ∪ {𝑧}))
96	93, 95	mpbir 234	. . . . . . . . . 10 ⊢ 𝑧 ∈ (𝑢 ∪ {𝑧})
97	92, 96	jctil 529	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (♯‘(𝑢 ∪ {𝑧})) = 𝐾))
98	63, 72, 97	elrabd 3647	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
99	98	ex 418	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
100	60, 99	biimtrid 245	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
101		eleq2 2850	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑣))
102		fveqeq2 6892	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → ((♯‘𝑥) = 𝐾 ↔ (♯‘𝑣) = 𝐾))
103	101, 102	anbi12d 644	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ((𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))
104	103	elrab 3645	. . . . . . 7 ⊢ (𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ↔ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))
105		fveqeq2 6892	. . . . . . . . 9 ⊢ (𝑥 = (𝑣 ∖ {𝑧}) → ((♯‘𝑥) = (𝐾 − 1) ↔ (♯‘(𝑣 ∖ {𝑧})) = (𝐾 − 1)))
106		elpwi 4564	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑣 ⊆ (𝐴 ∪ {𝑧}))
107	106	ad2antrl 741	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑣 ⊆ (𝐴 ∪ {𝑧}))
108	107, 19	sseqtrdi 3971	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑣 ⊆ ({𝑧} ∪ 𝐴))
109		ssundif 4443	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ ({𝑧} ∪ 𝐴) ↔ (𝑣 ∖ {𝑧}) ⊆ 𝐴)
110	108, 109	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ⊆ 𝐴)
111		vex 3455	. . . . . . . . . . . 12 ⊢ 𝑣 ∈ V
112	111	difexi 5292	. . . . . . . . . . 11 ⊢ (𝑣 ∖ {𝑧}) ∈ V
113	112	elpw 4561	. . . . . . . . . 10 ⊢ ((𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴 ↔ (𝑣 ∖ {𝑧}) ⊆ 𝐴)
114	110, 113	sylibr 237	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴)
115	46	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝐴 ∈ Fin)
116	115, 110	ssfid 9253	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ Fin)
117		hashcl 14493	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∖ {𝑧}) ∈ Fin → (♯‘(𝑣 ∖ {𝑧})) ∈ ℕ₀)
118	116, 117	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) ∈ ℕ₀)
119	118	nn0cnd 12662	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) ∈ ℂ)
120		pncan 11556	. . . . . . . . . . 11 ⊢ (((♯‘(𝑣 ∖ {𝑧})) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (♯‘(𝑣 ∖ {𝑧})))
121	119, 89, 120	sylancl 598	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (♯‘(𝑣 ∖ {𝑧})))
122		undif1 4430	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = (𝑣 ∪ {𝑧})
123		simprrl 793	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑧 ∈ 𝑣)
124	123	snssd 4747	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ⊆ 𝑣)
125		ssequn2 4135	. . . . . . . . . . . . . . 15 ⊢ ({𝑧} ⊆ 𝑣 ↔ (𝑣 ∪ {𝑧}) = 𝑣)
126	124, 125	sylib 221	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∪ {𝑧}) = 𝑣)
127	122, 126	eqtrid 2808	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = 𝑣)
128	127	fveq2d 6887	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = (♯‘𝑣))
129	51	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ∈ Fin)
130		disjdifr 4427	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅
131	130	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅)
132		hashun 14519	. . . . . . . . . . . . . 14 ⊢ (((𝑣 ∖ {𝑧}) ∈ Fin ∧ {𝑧} ∈ Fin ∧ ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})))
133	116, 129, 131, 132	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})))
134	84	oveq2i 7429	. . . . . . . . . . . . 13 ⊢ ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + 1)
135	133, 134	eqtrdi 2812	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + 1))
136		simprrr 794	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘𝑣) = 𝐾)
137	128, 135, 136	3eqtr3d 2804	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((♯‘(𝑣 ∖ {𝑧})) + 1) = 𝐾)
138	137	oveq1d 7433	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (𝐾 − 1))
139	121, 138	eqtr3d 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) = (𝐾 − 1))
140	105, 114, 139	elrabd 3647	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})
141	140	ex 418	. . . . . . 7 ⊢ (𝜑 → ((𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
142	104, 141	biimtrid 245	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
143	60, 104	anbi12i 640	. . . . . . 7 ⊢ ((𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∧ 𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) ↔ ((𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))))
144		simp3rl 1265	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑧 ∈ 𝑣)
145	144	snssd 4747	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ⊆ 𝑣)
146		incom 4155	. . . . . . . . . . . 12 ⊢ ({𝑧} ∩ 𝑢) = (𝑢 ∩ {𝑧})
147	79	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑢 ∩ {𝑧}) = ∅)
148	146, 147	eqtrid 2808	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ({𝑧} ∩ 𝑢) = ∅)
149		uneqdifeq 4448	. . . . . . . . . . 11 ⊢ (({𝑧} ⊆ 𝑣 ∧ ({𝑧} ∩ 𝑢) = ∅) → (({𝑧} ∪ 𝑢) = 𝑣 ↔ (𝑣 ∖ {𝑧}) = 𝑢))
150	145, 148, 149	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (({𝑧} ∪ 𝑢) = 𝑣 ↔ (𝑣 ∖ {𝑧}) = 𝑢))
151	150	bicomd 226	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) = 𝑢 ↔ ({𝑧} ∪ 𝑢) = 𝑣))
152		eqcom 2768	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ∖ {𝑧}) ↔ (𝑣 ∖ {𝑧}) = 𝑢)
153		eqcom 2768	. . . . . . . . . 10 ⊢ (𝑣 = (𝑢 ∪ {𝑧}) ↔ (𝑢 ∪ {𝑧}) = 𝑣)
154		uncom 4105	. . . . . . . . . . 11 ⊢ (𝑢 ∪ {𝑧}) = ({𝑧} ∪ 𝑢)
155	154	eqeq1i 2766	. . . . . . . . . 10 ⊢ ((𝑢 ∪ {𝑧}) = 𝑣 ↔ ({𝑧} ∪ 𝑢) = 𝑣)
156	153, 155	bitri 278	. . . . . . . . 9 ⊢ (𝑣 = (𝑢 ∪ {𝑧}) ↔ ({𝑧} ∪ 𝑢) = 𝑣)
157	151, 152, 156	3bitr4g 317	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧})))
158	157	3expib 1140	. . . . . . 7 ⊢ (𝜑 → (((𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧}))))
159	143, 158	biimtrid 245	. . . . . 6 ⊢ (𝜑 → ((𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∧ 𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧}))))
160	50, 58, 100, 142, 159	en3d 9009	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
161		ssrab2 4028	. . . . . . 7 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ⊆ 𝒫 𝐴
162		ssfi 9181	. . . . . . 7 ⊢ ((𝒫 𝐴 ∈ Fin ∧ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ⊆ 𝒫 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ Fin)
163	48, 161, 162	sylancl 598	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ Fin)
164		hashen 14484	. . . . . 6 ⊢ (({𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ Fin ∧ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin) → ((♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) ↔ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
165	163, 58, 164	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) ↔ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
166	160, 165	mpbird 260	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
167	45, 166	eqtrd 2796	. . 3 ⊢ (𝜑 → ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
168	37, 167	oveq12d 7436	. 2 ⊢ (𝜑 → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = ((♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})))
169	51	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑧} ∈ Fin)
170		disjsn 4672	. . . . . . 7 ⊢ ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝐴)
171	13, 170	sylibr 237	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
172		hashun 14519	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝐴 ∩ {𝑧}) = ∅) → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + (♯‘{𝑧})))
173	46, 169, 171, 172	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + (♯‘{𝑧})))
174	84	oveq2i 7429	. . . . 5 ⊢ ((♯‘𝐴) + (♯‘{𝑧})) = ((♯‘𝐴) + 1)
175	173, 174	eqtrdi 2812	. . . 4 ⊢ (𝜑 → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + 1))
176	175	oveq1d 7433	. . 3 ⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (((♯‘𝐴) + 1)C𝐾))
177		hashcl 14493	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
178	46, 177	syl 18	. . . 4 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
179		bcpasc 14458	. . . 4 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ 𝐾 ∈ ℤ) → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = (((♯‘𝐴) + 1)C𝐾))
180	178, 7, 179	syl2anc 596	. . 3 ⊢ (𝜑 → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = (((♯‘𝐴) + 1)C𝐾))
181	176, 180	eqtr4d 2799	. 2 ⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))))
182		pm2.1 910	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥)
183	182	biantrur 540	. . . . . . 7 ⊢ ((♯‘𝑥) = 𝐾 ↔ ((¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾))
184		andir 1026	. . . . . . 7 ⊢ (((¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾) ↔ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)))
185	183, 184	bitri 278	. . . . . 6 ⊢ ((♯‘𝑥) = 𝐾 ↔ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)))
186	185	rabbii 3418	. . . . 5 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾} = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))}
187		unrab 4261	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))}
188	186, 187	eqtr4i 2787	. . . 4 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾} = ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
189	188	fveq2i 6886	. . 3 ⊢ (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}) = (♯‘({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
190		ssrab2 4028	. . . . 5 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})
191		ssfi 9181	. . . . 5 ⊢ ((𝒫 (𝐴 ∪ {𝑧}) ∈ Fin ∧ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})) → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
192	55, 190, 191	sylancl 598	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
193		inrab 4262	. . . . . 6 ⊢ ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))}
194		simprl 783	. . . . . . . . 9 ⊢ (((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)) → 𝑧 ∈ 𝑥)
195		simpll 779	. . . . . . . . 9 ⊢ (((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)) → ¬ 𝑧 ∈ 𝑥)
196	194, 195	pm2.65i 196	. . . . . . . 8 ⊢ ¬ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))
197	196	rgenw 3081	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ¬ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))
198		rabeq0 4338	. . . . . . 7 ⊢ ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))} = ∅ ↔ ∀𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ¬ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)))
199	197, 198	mpbir 234	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))} = ∅
200	193, 199	eqtri 2784	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) = ∅
201	200	a1i 11	. . . 4 ⊢ (𝜑 → ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) = ∅)
202		hashun 14519	. . . 4 ⊢ (({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin ∧ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin ∧ ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) = ∅) → (♯‘({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})) = ((♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})))
203	192, 58, 201, 202	syl3anc 1398	. . 3 ⊢ (𝜑 → (♯‘({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})) = ((♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})))
204	189, 203	eqtrid 2808	. 2 ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}) = ((♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})))
205	168, 181, 204	3eqtr4d 2806	1 ⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3077 {crab 3413 Vcvv 3451 ∖ cdif 3896 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 𝒫 cpw 4557 {csn 4584 class class class wbr 5103 ‘cfv 6537 (class class class)co 7418 ≈ cen 8963 Fincfn 8966 ℂcc 11191 1c1 11194 + caddc 11196 − cmin 11534 ℕ₀cn0 12599 ℤcz 12686 Ccbc 14439 ♯chash 14467

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-oadd 8473 df-er 8710 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-dju 9975 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-n0 12600 df-z 12687 df-uz 12959 df-rp 13114 df-fz 13633 df-seq 14138 df-fac 14411 df-bc 14440 df-hash 14468

This theorem is used by: hashbc 14591

W3C validator