Theorem pgindnf 50477

Description: Version of pgind 50478 with extraneous not-free requirements. (Contributed by Emmett Weisz, 27-May-2024.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pgindnf.1	⊢ Ⅎ𝑥𝜑
pgindnf.2	⊢ Ⅎ𝑦𝜑
pgindnf.3	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
pgindnf.4	⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃))
pgindnf.5	⊢ (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))

Assertion

Ref	Expression
pgindnf	⊢ (𝜑 → (𝐴 ∈ Pg → 𝜃))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝜒,𝑥   𝜓,𝑦   𝜃,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥)   𝐴(𝑥)

Proof of Theorem pgindnf

Dummy variables 𝑎 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pg 50471	. 2 ⊢ Pg = setrecs((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)))
2		pgindnf.4	. 2 ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃))
3		pgindnf.1	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
4		nfv 1944	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 𝜒
5	3, 4	nfan 1929	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑦 ∈ 𝑧 𝜒)
6		pgindlem 50476	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st ‘𝑥) ∪ (2nd ‘𝑥)) ⊆ 𝑧)
7	6	sseld 3937	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥)) → 𝑦 ∈ 𝑧))
8	7	imim1d 83	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((𝑦 ∈ 𝑧 → 𝜒) → (𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥)) → 𝜒)))
9	8	ralimdv2 3174	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (∀𝑦 ∈ 𝑧 𝜒 → ∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒))
10		pgindnf.5	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
11	10	19.21bi 2225	. . . . . . . 8 ⊢ (𝜑 → (∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
12	9, 11	sylan9r 517	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)) → (∀𝑦 ∈ 𝑧 𝜒 → 𝜓))
13	12	impancom 456	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑧 𝜒) → (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → 𝜓))
14	5, 13	ralrimi 3263	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑧 𝜒) → ∀𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)𝜓)
15		pgindnf.3	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
16		vex 3459	. . . . . . . . 9 ⊢ 𝑧 ∈ V
17		pweq 4577	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑧 → 𝒫 𝑎 = 𝒫 𝑧)
18	17	sqxpeqd 5695	. . . . . . . . . 10 ⊢ (𝑎 = 𝑧 → (𝒫 𝑎 × 𝒫 𝑎) = (𝒫 𝑧 × 𝒫 𝑧))
19		eqid 2763	. . . . . . . . . 10 ⊢ (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)) = (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))
20		vpwex 5350	. . . . . . . . . . 11 ⊢ 𝒫 𝑧 ∈ V
21	20, 20	xpex 7753	. . . . . . . . . 10 ⊢ (𝒫 𝑧 × 𝒫 𝑧) ∈ V
22	18, 19, 21	fvmpt 6991	. . . . . . . . 9 ⊢ (𝑧 ∈ V → ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧) = (𝒫 𝑧 × 𝒫 𝑧))
23	16, 22	ax-mp 5	. . . . . . . 8 ⊢ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧) = (𝒫 𝑧 × 𝒫 𝑧)
24	23	eqcomi 2772	. . . . . . 7 ⊢ (𝒫 𝑧 × 𝒫 𝑧) = ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)
25	24	a1i 11	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝒫 𝑧 × 𝒫 𝑧) = ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧))
26	15, 25	cbvralv2 3900	. . . . 5 ⊢ (∀𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)𝜓 ↔ ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒)
27	14, 26	sylib 221	. . . 4 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑧 𝜒) → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒)
28	27	ex 417	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑧 𝜒 → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒))
29	28	alrimiv 1957	. 2 ⊢ (𝜑 → ∀𝑧(∀𝑦 ∈ 𝑧 𝜒 → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒))
30	1, 2, 29	setis 50459	1 ⊢ (𝜑 → (𝐴 ∈ Pg → 𝜃))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570  Ⅎwnf 1813   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∪ cun 3904  𝒫 cpw 4563   ↦ cmpt 5193   × cxp 5661  ‘cfv 6538  1^st c1st 7985  2^nd c2nd 7986  Pgcpg 50470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fv 6546  df-1st 7987  df-2nd 7988  df-setrecs 50445  df-pg 50471

This theorem is referenced by:  pgind  50478