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Theorem hashbclem 14349

Description: Lemma for hashbc 14350: 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 7365	. . . . . 6 ⊢ (𝑗 = 𝐾 → ((♯‘𝐴)C𝑗) = ((♯‘𝐴)C𝐾))
2		eqeq2 2748	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑥) = 𝐾))
3	2	rabbidv 3415	. . . . . . 7 ⊢ (𝑗 = 𝐾 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})
4	3	fveq2d 6846	. . . . . 6 ⊢ (𝑗 = 𝐾 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
5	1, 4	eqeq12d 2752	. . . . 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 3582	. . . 4 ⊢ (𝜑 → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
9		ssun1 4132	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑧})
10	9	sspwi 4572	. . . . . . . . . . . 12 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ {𝑧})
11	10	sseli 3940	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}))
12	11	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}))
13		hashbc.2	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
14		elpwi 4567	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
15	14	ssneld 3946	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝐴 → (¬ 𝑧 ∈ 𝐴 → ¬ 𝑧 ∈ 𝑥))
16	13, 15	mpan9 507	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → ¬ 𝑧 ∈ 𝑥)
17	12, 16	jca 512	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥))
18		elpwi 4567	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑥 ⊆ (𝐴 ∪ {𝑧}))
19		uncom 4113	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∪ {𝑧}) = ({𝑧} ∪ 𝐴)
20	18, 19	sseqtrdi 3994	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑥 ⊆ ({𝑧} ∪ 𝐴))
21	20	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ⊆ ({𝑧} ∪ 𝐴))
22		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → ¬ 𝑧 ∈ 𝑥)
23		disjsn 4672	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑥)
24	22, 23	sylibr 233	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → (𝑥 ∩ {𝑧}) = ∅)
25		disjssun 4427	. . . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ⊆ 𝐴)
28		vex 3449	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
29	28	elpw 4564	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
30	27, 29	sylibr 233	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ∈ 𝒫 𝐴)
31	30	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥)) → 𝑥 ∈ 𝒫 𝐴)
32	17, 31	impbida 799	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥)))
33	32	anbi1d 630	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (♯‘𝑥) = 𝐾) ↔ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾)))
34		anass 469	. . . . . . 7 ⊢ (((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾) ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)))
35	33, 34	bitrdi 286	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))))
36	35	rabbidva2 3409	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾} = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
37	36	fveq2d 6846	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
38	8, 37	eqtrd 2776	. . 3 ⊢ (𝜑 → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
39		oveq2 7365	. . . . . 6 ⊢ (𝑗 = (𝐾 − 1) → ((♯‘𝐴)C𝑗) = ((♯‘𝐴)C(𝐾 − 1)))
40		eqeq2 2748	. . . . . . . 8 ⊢ (𝑗 = (𝐾 − 1) → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑥) = (𝐾 − 1)))
41	40	rabbidv 3415	. . . . . . 7 ⊢ (𝑗 = (𝐾 − 1) → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})
42	41	fveq2d 6846	. . . . . 6 ⊢ (𝑗 = (𝐾 − 1) → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
43	39, 42	eqeq12d 2752	. . . . 5 ⊢ (𝑗 = (𝐾 − 1) → (((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})))
44		peano2zm 12546	. . . . . 6 ⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ)
45	7, 44	syl 17	. . . . 5 ⊢ (𝜑 → (𝐾 − 1) ∈ ℤ)
46	43, 6, 45	rspcdva 3582	. . . 4 ⊢ (𝜑 → ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
47		hashbc.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Fin)
48		pwfi 9122	. . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
49	47, 48	sylib 217	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝐴 ∈ Fin)
50		rabexg 5288	. . . . . . 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 8988	. . . . . . . . 9 ⊢ {𝑧} ∈ Fin
53		unfi 9116	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
54	47, 52, 53	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
55		pwfi 9122	. . . . . . . 8 ⊢ ((𝐴 ∪ {𝑧}) ∈ Fin ↔ 𝒫 (𝐴 ∪ {𝑧}) ∈ Fin)
56	54, 55	sylib 217	. . . . . . 7 ⊢ (𝜑 → 𝒫 (𝐴 ∪ {𝑧}) ∈ Fin)
57		ssrab2 4037	. . . . . . 7 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})
58		ssfi 9117	. . . . . . 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 6851	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((♯‘𝑥) = (𝐾 − 1) ↔ (♯‘𝑢) = (𝐾 − 1)))
61	60	elrab 3645	. . . . . . 7 ⊢ (𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ↔ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)))
62		eleq2 2826	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ∪ {𝑧}) → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ (𝑢 ∪ {𝑧})))
63		fveqeq2 6851	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ∪ {𝑧}) → ((♯‘𝑥) = 𝐾 ↔ (♯‘(𝑢 ∪ {𝑧})) = 𝐾))
64	62, 63	anbi12d 631	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ∪ {𝑧}) → ((𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (♯‘(𝑢 ∪ {𝑧})) = 𝐾)))
65		elpwi 4567	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝒫 𝐴 → 𝑢 ⊆ 𝐴)
66	65	ad2antrl 726	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝑢 ⊆ 𝐴)
67		unss1 4139	. . . . . . . . . . 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 3449	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
70		vsnex 5386	. . . . . . . . . . . 12 ⊢ {𝑧} ∈ V
71	69, 70	unex 7680	. . . . . . . . . . 11 ⊢ (𝑢 ∪ {𝑧}) ∈ V
72	71	elpw 4564	. . . . . . . . . 10 ⊢ ((𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}) ↔ (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
73	68, 72	sylibr 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}))
74	47	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐴 ∈ Fin)
75	74, 66	ssfid 9211	. . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ¬ 𝑧 ∈ 𝐴)
78	66, 77	ssneldd 3947	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ¬ 𝑧 ∈ 𝑢)
79		disjsn 4672	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑢)
80	78, 79	sylibr 233	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∩ {𝑧}) = ∅)
81		hashun 14282	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝑢 ∩ {𝑧}) = ∅) → (♯‘(𝑢 ∪ {𝑧})) = ((♯‘𝑢) + (♯‘{𝑧})))
82	75, 76, 80, 81	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘(𝑢 ∪ {𝑧})) = ((♯‘𝑢) + (♯‘{𝑧})))
83		simprr 771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘𝑢) = (𝐾 − 1))
84		hashsng 14269	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → (♯‘{𝑧}) = 1)
85	84	elv 3451	. . . . . . . . . . . . 13 ⊢ (♯‘{𝑧}) = 1
86	85	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘{𝑧}) = 1)
87	83, 86	oveq12d 7375	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ((♯‘𝑢) + (♯‘{𝑧})) = ((𝐾 − 1) + 1))
88	7	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℤ)
89	88	zcnd 12608	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℂ)
90		ax-1cn 11109	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
91		npcan 11410	. . . . . . . . . . . 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 2780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘(𝑢 ∪ {𝑧})) = 𝐾)
94		ssun2 4133	. . . . . . . . . . 11 ⊢ {𝑧} ⊆ (𝑢 ∪ {𝑧})
95		vex 3449	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
96	95	snss 4746	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑢 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝑢 ∪ {𝑧}))
97	94, 96	mpbir 230	. . . . . . . . . 10 ⊢ 𝑧 ∈ (𝑢 ∪ {𝑧})
98	93, 97	jctil 520	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (♯‘(𝑢 ∪ {𝑧})) = 𝐾))
99	64, 73, 98	elrabd 3647	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
100	99	ex 413	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
101	61, 100	biimtrid 241	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
102		eleq2 2826	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑣))
103		fveqeq2 6851	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → ((♯‘𝑥) = 𝐾 ↔ (♯‘𝑣) = 𝐾))
104	102, 103	anbi12d 631	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ((𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))
105	104	elrab 3645	. . . . . . 7 ⊢ (𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ↔ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)))
106		fveqeq2 6851	. . . . . . . . 9 ⊢ (𝑥 = (𝑣 ∖ {𝑧}) → ((♯‘𝑥) = (𝐾 − 1) ↔ (♯‘(𝑣 ∖ {𝑧})) = (𝐾 − 1)))
107		elpwi 4567	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑣 ⊆ (𝐴 ∪ {𝑧}))
108	107	ad2antrl 726	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑣 ⊆ (𝐴 ∪ {𝑧}))
109	108, 19	sseqtrdi 3994	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑣 ⊆ ({𝑧} ∪ 𝐴))
110		ssundif 4445	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ ({𝑧} ∪ 𝐴) ↔ (𝑣 ∖ {𝑧}) ⊆ 𝐴)
111	109, 110	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ⊆ 𝐴)
112		vex 3449	. . . . . . . . . . . 12 ⊢ 𝑣 ∈ V
113	112	difexi 5285	. . . . . . . . . . 11 ⊢ (𝑣 ∖ {𝑧}) ∈ V
114	113	elpw 4564	. . . . . . . . . 10 ⊢ ((𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴 ↔ (𝑣 ∖ {𝑧}) ⊆ 𝐴)
115	111, 114	sylibr 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴)
116	47	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝐴 ∈ Fin)
117	116, 111	ssfid 9211	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ Fin)
118		hashcl 14256	. . . . . . . . . . . . 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 12475	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) ∈ ℂ)
121		pncan 11407	. . . . . . . . . . 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 4435	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = (𝑣 ∪ {𝑧})
124		simprrl 779	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑧 ∈ 𝑣)
125	124	snssd 4769	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ⊆ 𝑣)
126		ssequn2 4143	. . . . . . . . . . . . . . 15 ⊢ ({𝑧} ⊆ 𝑣 ↔ (𝑣 ∪ {𝑧}) = 𝑣)
127	125, 126	sylib 217	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∪ {𝑧}) = 𝑣)
128	123, 127	eqtrid 2788	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = 𝑣)
129	128	fveq2d 6846	. . . . . . . . . . . 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 4432	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅
132	131	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅)
133		hashun 14282	. . . . . . . . . . . . . 14 ⊢ (((𝑣 ∖ {𝑧}) ∈ Fin ∧ {𝑧} ∈ Fin ∧ ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})))
134	117, 130, 132, 133	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})))
135	85	oveq2i 7368	. . . . . . . . . . . . 13 ⊢ ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + 1)
136	134, 135	eqtrdi 2792	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + 1))
137		simprrr 780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘𝑣) = 𝐾)
138	129, 136, 137	3eqtr3d 2784	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((♯‘(𝑣 ∖ {𝑧})) + 1) = 𝐾)
139	138	oveq1d 7372	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (𝐾 − 1))
140	122, 139	eqtr3d 2778	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) = (𝐾 − 1))
141	106, 115, 140	elrabd 3647	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})
142	141	ex 413	. . . . . . 7 ⊢ (𝜑 → ((𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾)) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
143	105, 142	biimtrid 241	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
144	61, 105	anbi12i 627	. . . . . . 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 1246	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑧 ∈ 𝑣)
146	145	snssd 4769	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ⊆ 𝑣)
147		incom 4161	. . . . . . . . . . . 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 2788	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ({𝑧} ∩ 𝑢) = ∅)
150		uneqdifeq 4450	. . . . . . . . . . 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 222	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) = 𝑢 ↔ ({𝑧} ∪ 𝑢) = 𝑣))
153		eqcom 2743	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ∖ {𝑧}) ↔ (𝑣 ∖ {𝑧}) = 𝑢)
154		eqcom 2743	. . . . . . . . . 10 ⊢ (𝑣 = (𝑢 ∪ {𝑧}) ↔ (𝑢 ∪ {𝑧}) = 𝑣)
155		uncom 4113	. . . . . . . . . . 11 ⊢ (𝑢 ∪ {𝑧}) = ({𝑧} ∪ 𝑢)
156	155	eqeq1i 2741	. . . . . . . . . 10 ⊢ ((𝑢 ∪ {𝑧}) = 𝑣 ↔ ({𝑧} ∪ 𝑢) = 𝑣)
157	154, 156	bitri 274	. . . . . . . . 9 ⊢ (𝑣 = (𝑢 ∪ {𝑧}) ↔ ({𝑧} ∪ 𝑢) = 𝑣)
158	152, 153, 157	3bitr4g 313	. . . . . . . 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 241	. . . . . 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 8929	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)})
162		ssrab2 4037	. . . . . . 7 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ⊆ 𝒫 𝐴
163		ssfi 9117	. . . . . . 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 14247	. . . . . 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 256	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
168	46, 167	eqtrd 2776	. . 3 ⊢ (𝜑 → ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)}))
169	38, 168	oveq12d 7375	. 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 4672	. . . . . . 7 ⊢ ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝐴)
172	13, 171	sylibr 233	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
173		hashun 14282	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝐴 ∩ {𝑧}) = ∅) → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + (♯‘{𝑧})))
174	47, 170, 172, 173	syl3anc 1371	. . . . 5 ⊢ (𝜑 → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + (♯‘{𝑧})))
175	85	oveq2i 7368	. . . . 5 ⊢ ((♯‘𝐴) + (♯‘{𝑧})) = ((♯‘𝐴) + 1)
176	174, 175	eqtrdi 2792	. . . 4 ⊢ (𝜑 → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + 1))
177	176	oveq1d 7372	. . 3 ⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (((♯‘𝐴) + 1)C𝐾))
178		hashcl 14256	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
179	47, 178	syl 17	. . . 4 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
180		bcpasc 14221	. . . 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 2779	. 2 ⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))))
183		pm2.1 895	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥)
184	183	biantrur 531	. . . . . . 7 ⊢ ((♯‘𝑥) = 𝐾 ↔ ((¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾))
185		andir 1007	. . . . . . 7 ⊢ (((¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥) ∧ (♯‘𝑥) = 𝐾) ↔ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)))
186	184, 185	bitri 274	. . . . . 6 ⊢ ((♯‘𝑥) = 𝐾 ↔ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)))
187	186	rabbii 3413	. . . . 5 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾} = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))}
188		unrab 4265	. . . . 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 2767	. . . 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 6845	. . 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 4037	. . . . 5 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})
192		ssfi 9117	. . . . 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 4266	. . . . . 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 769	. . . . . . . . 9 ⊢ (((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)) → 𝑧 ∈ 𝑥)
196		simpll 765	. . . . . . . . 9 ⊢ (((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾)) → ¬ 𝑧 ∈ 𝑥)
197	195, 196	pm2.65i 193	. . . . . . . 8 ⊢ ¬ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))
198	197	rgenw 3068	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ¬ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))
199		rabeq0 4344	. . . . . . 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 230	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (♯‘𝑥) = 𝐾))} = ∅
201	194, 200	eqtri 2764	. . . . 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 14282	. . . 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 1371	. . 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 2788	. 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 2786	1 ⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 {crab 3407 Vcvv 3445 ∖ cdif 3907 ∪ cun 3908 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 𝒫 cpw 4560 {csn 4586 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ≈ cen 8880 Fincfn 8883 ℂcc 11049 1c1 11052 + caddc 11054 − cmin 11385 ℕ₀cn0 12413 ℤcz 12499 Ccbc 14202 ♯chash 14230

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-seq 13907 df-fac 14174 df-bc 14203 df-hash 14231

This theorem is referenced by: hashbc 14350

W3C validator