Theorem pgindnf 50206

Description: Version of pgind 50207 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 50200	. 2 ⊢ Pg = setrecs((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)))
2		pgindnf.4	. 2 ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃))
3		pgindnf.1	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
4		nfv 1921	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 𝜒
5	3, 4	nfan 1906	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑦 ∈ 𝑧 𝜒)
6		pgindlem 50205	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st ‘𝑥) ∪ (2nd ‘𝑥)) ⊆ 𝑧)
7	6	sseld 3914	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥)) → 𝑦 ∈ 𝑧))
8	7	imim1d 82	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((𝑦 ∈ 𝑧 → 𝜒) → (𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥)) → 𝜒)))
9	8	ralimdv2 3148	. . . . . . . 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 2201	. . . . . . . 8 ⊢ (𝜑 → (∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
12	9, 11	sylan9r 513	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)) → (∀𝑦 ∈ 𝑧 𝜒 → 𝜓))
13	12	impancom 452	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑧 𝜒) → (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → 𝜓))
14	5, 13	ralrimi 3237	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑧 𝜒) → ∀𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)𝜓)
15		pgindnf.3	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
16		vex 3435	. . . . . . . . 9 ⊢ 𝑧 ∈ V
17		pweq 4543	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑧 → 𝒫 𝑎 = 𝒫 𝑧)
18	17	sqxpeqd 5650	. . . . . . . . . 10 ⊢ (𝑎 = 𝑧 → (𝒫 𝑎 × 𝒫 𝑎) = (𝒫 𝑧 × 𝒫 𝑧))
19		eqid 2739	. . . . . . . . . 10 ⊢ (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)) = (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))
20		vpwex 5306	. . . . . . . . . . 11 ⊢ 𝒫 𝑧 ∈ V
21	20, 20	xpex 7696	. . . . . . . . . 10 ⊢ (𝒫 𝑧 × 𝒫 𝑧) ∈ V
22	18, 19, 21	fvmpt 6935	. . . . . . . . 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 2748	. . . . . . 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 3877	. . . . 5 ⊢ (∀𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)𝜓 ↔ ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒)
27	14, 26	sylib 219	. . . 4 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑧 𝜒) → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒)
28	27	ex 413	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑧 𝜒 → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒))
29	28	alrimiv 1934	. 2 ⊢ (𝜑 → ∀𝑧(∀𝑦 ∈ 𝑧 𝜒 → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒))
30	1, 2, 29	setis 50188	1 ⊢ (𝜑 → (𝐴 ∈ Pg → 𝜃))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ∪ cun 3881  𝒫 cpw 4529   ↦ cmpt 5153   × cxp 5616  ‘cfv 6485  1^st c1st 7929  2^nd c2nd 7930  Pgcpg 50199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fv 6493  df-1st 7931  df-2nd 7932  df-setrecs 50174  df-pg 50200

This theorem is referenced by:  pgind  50207