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Theorem hashbclem 14470

Description: Lemma for hashbc 14471: 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 7413	. . . . . 6 ⊢ (𝑗 = 𝐾 → ((♯‘𝐴)C𝑗) = ((♯‘𝐴)C𝐾))
2		eqeq2 2747	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑥) = 𝐾))
3	2	rabbidv 3423	. . . . . . 7 ⊢ (𝑗 = 𝐾 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})
4	3	fveq2d 6880	. . . . . 6 ⊢ (𝑗 = 𝐾 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
5	1, 4	eqeq12d 2751	. . . . 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 3602	. . . 4 ⊢ (𝜑 → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
9		ssun1 4153	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑧})
10	9	sspwi 4587	. . . . . . . . . . . 12 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ {𝑧})
11	10	sseli 3954	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}))
12	11	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}))
13		hashbc.2	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
14		elpwi 4582	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
15	14	ssneld 3960	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝐴 → (¬ 𝑧 ∈ 𝐴 → ¬ 𝑧 ∈ 𝑥))
16	13, 15	mpan9 506	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → ¬ 𝑧 ∈ 𝑥)
17	12, 16	jca 511	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥))
18		elpwi 4582	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑥 ⊆ (𝐴 ∪ {𝑧}))
19		uncom 4133	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∪ {𝑧}) = ({𝑧} ∪ 𝐴)
20	18, 19	sseqtrdi 3999	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑥 ⊆ ({𝑧} ∪ 𝐴))
21	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ⊆ ({𝑧} ∪ 𝐴))
22		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → ¬ 𝑧 ∈ 𝑥)
23		disjsn 4687	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑥)
24	22, 23	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → (𝑥 ∩ {𝑧}) = ∅)
25		disjssun 4443	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ {𝑧}) = ∅ → (𝑥 ⊆ ({𝑧} ∪ 𝐴) ↔ 𝑥 ⊆ 𝐴))
26	24, 25	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → (𝑥 ⊆ ({𝑧} ∪ 𝐴) ↔ 𝑥 ⊆ 𝐴))
27	21, 26	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ⊆ 𝐴)
28		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
29	28	elpw 4579	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
30	27, 29	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ∈ 𝒫 𝐴)
31	30	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥)) → 𝑥 ∈ 𝒫 𝐴)
32	17, 31	impbida 800	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥)))
33	32	anbi1d 631	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (♯‘𝑥) = 𝐾) ↔ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾)))
34		anass 468	. . . . . . 7 ⊢ (((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾) ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)))
35	33, 34	bitrdi 287	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))))
36	35	rabbidva2 3417	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾} = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
37	36	fveq2d 6880	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
38	8, 37	eqtrd 2770	. . 3 ⊢ (𝜑 → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
39		oveq2 7413	. . . . . 6 ⊢ (𝑗 = (𝐾 − 1) → ((♯‘𝐴)C𝑗) = ((♯‘𝐴)C(𝐾 − 1)))
40		eqeq2 2747	. . . . . . . 8 ⊢ (𝑗 = (𝐾 − 1) → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑥) = (𝐾 − 1)))
41	40	rabbidv 3423	. . . . . . 7 ⊢ (𝑗 = (𝐾 − 1) → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})
42	41	fveq2d 6880	. . . . . 6 ⊢ (𝑗 = (𝐾 − 1) → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
43	39, 42	eqeq12d 2751	. . . . 5 ⊢ (𝑗 = (𝐾 − 1) → (((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})))
44		peano2zm 12635	. . . . . 6 ⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ)
45	7, 44	syl 17	. . . . 5 ⊢ (𝜑 → (𝐾 − 1) ∈ ℤ)
46	43, 6, 45	rspcdva 3602	. . . 4 ⊢ (𝜑 → ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
47		hashbc.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Fin)
48		pwfi 9329	. . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
49	47, 48	sylib 218	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝐴 ∈ Fin)
50		rabexg 5307	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ Fin → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ V)
51	49, 50	syl 17	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ V)
52		snfi 9057	. . . . . . . . 9 ⊢ {𝑧} ∈ Fin
53		unfi 9185	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
54	47, 52, 53	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
55		pwfi 9329	. . . . . . . 8 ⊢ ((𝐴 ∪ {𝑧}) ∈ Fin ↔ 𝒫 (𝐴 ∪ {𝑧}) ∈ Fin)
56	54, 55	sylib 218	. . . . . . 7 ⊢ (𝜑 → 𝒫 (𝐴 ∪ {𝑧}) ∈ Fin)
57		ssrab2 4055	. . . . . . 7 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})
58		ssfi 9187	. . . . . . 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)
59	56, 57, 58	sylancl 586	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
60		fveqeq2 6885	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((♯‘𝑥) = (𝐾 − 1) ↔ (♯‘𝑢) = (𝐾 − 1)))
61	60	elrab 3671	. . . . . . 7 ⊢ (𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ↔ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)))
62		eleq2 2823	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ∪ {𝑧}) → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ (𝑢 ∪ {𝑧})))
63		fveqeq2 6885	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ∪ {𝑧}) → ((♯‘𝑥) = 𝐾 ↔ (♯‘(𝑢 ∪ {𝑧})) = 𝐾))
64	62, 63	anbi12d 632	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ∪ {𝑧}) → ((𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (♯‘(𝑢 ∪ {𝑧})) = 𝐾)))
65		elpwi 4582	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝒫 𝐴 → 𝑢 ⊆ 𝐴)
66	65	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝑢 ⊆ 𝐴)
67		unss1 4160	. . . . . . . . . . 11 ⊢ (𝑢 ⊆ 𝐴 → (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
68	66, 67	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
69		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
70		vsnex 5404	. . . . . . . . . . . 12 ⊢ {𝑧} ∈ V
71	69, 70	unex 7738	. . . . . . . . . . 11 ⊢ (𝑢 ∪ {𝑧}) ∈ V
72	71	elpw 4579	. . . . . . . . . 10 ⊢ ((𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}) ↔ (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
73	68, 72	sylibr 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}))
74	47	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐴 ∈ Fin)
75	74, 66	ssfid 9273	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝑢 ∈ Fin)
76	52	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → {𝑧} ∈ Fin)
77	13	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ¬ 𝑧 ∈ 𝐴)
78	66, 77	ssneldd 3961	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ¬ 𝑧 ∈ 𝑢)
79		disjsn 4687	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑢)
80	78, 79	sylibr 234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∩ {𝑧}) = ∅)
81		hashun 14400	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝑢 ∩ {𝑧}) = ∅) → (♯‘(𝑢 ∪ {𝑧})) = ((♯‘𝑢) + (♯‘{𝑧})))
82	75, 76, 80, 81	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘(𝑢 ∪ {𝑧})) = ((♯‘𝑢) + (♯‘{𝑧})))
83		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘𝑢) = (𝐾 − 1))
84		hashsng 14387	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → (♯‘{𝑧}) = 1)
85	84	elv 3464	. . . . . . . . . . . . 13 ⊢ (♯‘{𝑧}) = 1
86	85	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘{𝑧}) = 1)
87	83, 86	oveq12d 7423	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ((♯‘𝑢) + (♯‘{𝑧})) = ((𝐾 − 1) + 1))
88	7	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℤ)
89	88	zcnd 12698	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℂ)
90		ax-1cn 11187	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
91		npcan 11491	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 − 1) + 1) = 𝐾)
92	89, 90, 91	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ((𝐾 − 1) + 1) = 𝐾)
93	82, 87, 92	3eqtrd 2774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘(𝑢 ∪ {𝑧})) = 𝐾)
94		ssun2 4154	. . . . . . . . . . 11 ⊢ {𝑧} ⊆ (𝑢 ∪ {𝑧})
95		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
96	95	snss 4761	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑢 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝑢 ∪ {𝑧}))
97	94, 96	mpbir 231	. . . . . . . . . 10 ⊢ 𝑧 ∈ (𝑢 ∪ {𝑧})
98	93, 97	jctil 519	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (♯‘(𝑢 ∪ {𝑧})) = 𝐾))
99	64, 73, 98	elrabd 3673	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
100	99	ex 412	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
101	61, 100	biimtrid 242	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
102		eleq2 2823	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑣))
103		fveqeq2 6885	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → ((♯‘𝑥) = 𝐾 ↔ (♯‘𝑣) = 𝐾))
104	102, 103	anbi12d 632	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ((𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))
105	104	elrab 3671	. . . . . . 7 ⊢ (𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ↔ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))
106		fveqeq2 6885	. . . . . . . . 9 ⊢ (𝑥 = (𝑣 ∖ {𝑧}) → ((♯‘𝑥) = (𝐾 − 1) ↔ (♯‘(𝑣 ∖ {𝑧})) = (𝐾 − 1)))
107		elpwi 4582	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑣 ⊆ (𝐴 ∪ {𝑧}))
108	107	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑣 ⊆ (𝐴 ∪ {𝑧}))
109	108, 19	sseqtrdi 3999	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑣 ⊆ ({𝑧} ∪ 𝐴))
110		ssundif 4463	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ ({𝑧} ∪ 𝐴) ↔ (𝑣 ∖ {𝑧}) ⊆ 𝐴)
111	109, 110	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ⊆ 𝐴)
112		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑣 ∈ V
113	112	difexi 5300	. . . . . . . . . . 11 ⊢ (𝑣 ∖ {𝑧}) ∈ V
114	113	elpw 4579	. . . . . . . . . 10 ⊢ ((𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴 ↔ (𝑣 ∖ {𝑧}) ⊆ 𝐴)
115	111, 114	sylibr 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴)
116	47	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝐴 ∈ Fin)
117	116, 111	ssfid 9273	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ Fin)
118		hashcl 14374	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∖ {𝑧}) ∈ Fin → (♯‘(𝑣 ∖ {𝑧})) ∈ ℕ₀)
119	117, 118	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) ∈ ℕ₀)
120	119	nn0cnd 12564	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) ∈ ℂ)
121		pncan 11488	. . . . . . . . . . 11 ⊢ (((♯‘(𝑣 ∖ {𝑧})) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (♯‘(𝑣 ∖ {𝑧})))
122	120, 90, 121	sylancl 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (♯‘(𝑣 ∖ {𝑧})))
123		undif1 4451	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = (𝑣 ∪ {𝑧})
124		simprrl 780	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑧 ∈ 𝑣)
125	124	snssd 4785	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ⊆ 𝑣)
126		ssequn2 4164	. . . . . . . . . . . . . . 15 ⊢ ({𝑧} ⊆ 𝑣 ↔ (𝑣 ∪ {𝑧}) = 𝑣)
127	125, 126	sylib 218	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∪ {𝑧}) = 𝑣)
128	123, 127	eqtrid 2782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = 𝑣)
129	128	fveq2d 6880	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = (♯‘𝑣))
130	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ∈ Fin)
131		disjdifr 4448	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅
132	131	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅)
133		hashun 14400	. . . . . . . . . . . . . 14 ⊢ (((𝑣 ∖ {𝑧}) ∈ Fin ∧ {𝑧} ∈ Fin ∧ ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})))
134	117, 130, 132, 133	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})))
135	85	oveq2i 7416	. . . . . . . . . . . . 13 ⊢ ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + 1)
136	134, 135	eqtrdi 2786	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + 1))
137		simprrr 781	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘𝑣) = 𝐾)
138	129, 136, 137	3eqtr3d 2778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((♯‘(𝑣 ∖ {𝑧})) + 1) = 𝐾)
139	138	oveq1d 7420	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (𝐾 − 1))
140	122, 139	eqtr3d 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) = (𝐾 − 1))
141	106, 115, 140	elrabd 3673	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})
142	141	ex 412	. . . . . . 7 ⊢ (𝜑 → ((𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
143	105, 142	biimtrid 242	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
144	61, 105	anbi12i 628	. . . . . . 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))))
145		simp3rl 1247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑧 ∈ 𝑣)
146	145	snssd 4785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ⊆ 𝑣)
147		incom 4184	. . . . . . . . . . . 12 ⊢ ({𝑧} ∩ 𝑢) = (𝑢 ∩ {𝑧})
148	80	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑢 ∩ {𝑧}) = ∅)
149	147, 148	eqtrid 2782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ({𝑧} ∩ 𝑢) = ∅)
150		uneqdifeq 4468	. . . . . . . . . . 11 ⊢ (({𝑧} ⊆ 𝑣 ∧ ({𝑧} ∩ 𝑢) = ∅) → (({𝑧} ∪ 𝑢) = 𝑣 ↔ (𝑣 ∖ {𝑧}) = 𝑢))
151	146, 149, 150	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (({𝑧} ∪ 𝑢) = 𝑣 ↔ (𝑣 ∖ {𝑧}) = 𝑢))
152	151	bicomd 223	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) = 𝑢 ↔ ({𝑧} ∪ 𝑢) = 𝑣))
153		eqcom 2742	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ∖ {𝑧}) ↔ (𝑣 ∖ {𝑧}) = 𝑢)
154		eqcom 2742	. . . . . . . . . 10 ⊢ (𝑣 = (𝑢 ∪ {𝑧}) ↔ (𝑢 ∪ {𝑧}) = 𝑣)
155		uncom 4133	. . . . . . . . . . 11 ⊢ (𝑢 ∪ {𝑧}) = ({𝑧} ∪ 𝑢)
156	155	eqeq1i 2740	. . . . . . . . . 10 ⊢ ((𝑢 ∪ {𝑧}) = 𝑣 ↔ ({𝑧} ∪ 𝑢) = 𝑣)
157	154, 156	bitri 275	. . . . . . . . 9 ⊢ (𝑣 = (𝑢 ∪ {𝑧}) ↔ ({𝑧} ∪ 𝑢) = 𝑣)
158	152, 153, 157	3bitr4g 314	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧})))
159	158	3expib 1122	. . . . . . 7 ⊢ (𝜑 → (((𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧}))))
160	144, 159	biimtrid 242	. . . . . 6 ⊢ (𝜑 → ((𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∧ 𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧}))))
161	51, 59, 101, 143, 160	en3d 9003	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
162		ssrab2 4055	. . . . . . 7 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ⊆ 𝒫 𝐴
163		ssfi 9187	. . . . . . 7 ⊢ ((𝒫 𝐴 ∈ Fin ∧ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ⊆ 𝒫 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ Fin)
164	49, 162, 163	sylancl 586	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ Fin)
165		hashen 14365	. . . . . 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)}))
166	164, 59, 165	syl2anc 584	. . . . 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)}))
167	161, 166	mpbird 257	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
168	46, 167	eqtrd 2770	. . 3 ⊢ (𝜑 → ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
169	38, 168	oveq12d 7423	. 2 ⊢ (𝜑 → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = ((♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})))
170	52	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑧} ∈ Fin)
171		disjsn 4687	. . . . . . 7 ⊢ ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝐴)
172	13, 171	sylibr 234	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
173		hashun 14400	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝐴 ∩ {𝑧}) = ∅) → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + (♯‘{𝑧})))
174	47, 170, 172, 173	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + (♯‘{𝑧})))
175	85	oveq2i 7416	. . . . 5 ⊢ ((♯‘𝐴) + (♯‘{𝑧})) = ((♯‘𝐴) + 1)
176	174, 175	eqtrdi 2786	. . . 4 ⊢ (𝜑 → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + 1))
177	176	oveq1d 7420	. . 3 ⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (((♯‘𝐴) + 1)C𝐾))
178		hashcl 14374	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
179	47, 178	syl 17	. . . 4 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
180		bcpasc 14339	. . . 4 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ 𝐾 ∈ ℤ) → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = (((♯‘𝐴) + 1)C𝐾))
181	179, 7, 180	syl2anc 584	. . 3 ⊢ (𝜑 → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = (((♯‘𝐴) + 1)C𝐾))
182	177, 181	eqtr4d 2773	. 2 ⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))))
183		pm2.1 896	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥)
184	183	biantrur 530	. . . . . . 7 ⊢ ((♯‘𝑥) = 𝐾 ↔ ((¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾))
185		andir 1010	. . . . . . 7 ⊢ (((¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾) ↔ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)))
186	184, 185	bitri 275	. . . . . 6 ⊢ ((♯‘𝑥) = 𝐾 ↔ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)))
187	186	rabbii 3421	. . . . 5 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾} = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))}
188		unrab 4290	. . . . 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))}
189	187, 188	eqtr4i 2761	. . . 4 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾} = ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
190	189	fveq2i 6879	. . 3 ⊢ (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}) = (♯‘({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
191		ssrab2 4055	. . . . 5 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})
192		ssfi 9187	. . . . 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)
193	56, 191, 192	sylancl 586	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
194		inrab 4291	. . . . . 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))}
195		simprl 770	. . . . . . . . 9 ⊢ (((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)) → 𝑧 ∈ 𝑥)
196		simpll 766	. . . . . . . . 9 ⊢ (((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)) → ¬ 𝑧 ∈ 𝑥)
197	195, 196	pm2.65i 194	. . . . . . . 8 ⊢ ¬ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))
198	197	rgenw 3055	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ¬ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))
199		rabeq0 4363	. . . . . . 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)))
200	198, 199	mpbir 231	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))} = ∅
201	194, 200	eqtri 2758	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) = ∅
202	201	a1i 11	. . . 4 ⊢ (𝜑 → ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) = ∅)
203		hashun 14400	. . . 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)})))
204	193, 59, 202, 203	syl3anc 1373	. . 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)})))
205	190, 204	eqtrid 2782	. 2 ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}) = ((♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})))
206	169, 182, 205	3eqtr4d 2780	1 ⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 {crab 3415 Vcvv 3459 ∖ cdif 3923 ∪ cun 3924 ∩ cin 3925 ⊆ wss 3926 ∅c0 4308 𝒫 cpw 4575 {csn 4601 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 ≈ cen 8956 Fincfn 8959 ℂcc 11127 1c1 11130 + caddc 11132 − cmin 11466 ℕ₀cn0 12501 ℤcz 12588 Ccbc 14320 ♯chash 14348

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-n0 12502 df-z 12589 df-uz 12853 df-rp 13009 df-fz 13525 df-seq 14020 df-fac 14292 df-bc 14321 df-hash 14349

This theorem is referenced by: hashbc 14471

W3C validator