Theorem pgindnf 49877

Description: Version of pgind 49878 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 49871	. 2 ⊢ Pg = setrecs((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)))
2		pgindnf.4	. 2 ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃))
3		pgindnf.1	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
4		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 𝜒
5	3, 4	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑦 ∈ 𝑧 𝜒)
6		pgindlem 49876	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st ‘𝑥) ∪ (2nd ‘𝑥)) ⊆ 𝑧)
7	6	sseld 3929	. . . . . . . . . 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 3142	. . . . . . . 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 2194	. . . . . . . 8 ⊢ (𝜑 → (∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))
12	9, 11	sylan9r 508	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)) → (∀𝑦 ∈ 𝑧 𝜒 → 𝜓))
13	12	impancom 451	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑧 𝜒) → (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → 𝜓))
14	5, 13	ralrimi 3231	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑧 𝜒) → ∀𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)𝜓)
15		pgindnf.3	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
16		vex 3441	. . . . . . . . 9 ⊢ 𝑧 ∈ V
17		pweq 4565	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑧 → 𝒫 𝑎 = 𝒫 𝑧)
18	17	sqxpeqd 5653	. . . . . . . . . 10 ⊢ (𝑎 = 𝑧 → (𝒫 𝑎 × 𝒫 𝑎) = (𝒫 𝑧 × 𝒫 𝑧))
19		eqid 2733	. . . . . . . . . 10 ⊢ (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)) = (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))
20		vpwex 5319	. . . . . . . . . . 11 ⊢ 𝒫 𝑧 ∈ V
21	20, 20	xpex 7695	. . . . . . . . . 10 ⊢ (𝒫 𝑧 × 𝒫 𝑧) ∈ V
22	18, 19, 21	fvmpt 6938	. . . . . . . . 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 2742	. . . . . . 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 3892	. . . . 5 ⊢ (∀𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)𝜓 ↔ ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒)
27	14, 26	sylib 218	. . . 4 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑧 𝜒) → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒)
28	27	ex 412	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑧 𝜒 → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒))
29	28	alrimiv 1928	. 2 ⊢ (𝜑 → ∀𝑧(∀𝑦 ∈ 𝑧 𝜒 → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒))
30	1, 2, 29	setis 49859	1 ⊢ (𝜑 → (𝐴 ∈ Pg → 𝜃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2113  ∀wral 3048  Vcvv 3437   ∪ cun 3896  𝒫 cpw 4551   ↦ cmpt 5176   × cxp 5619  ‘cfv 6489  1^st c1st 7928  2^nd c2nd 7929  Pgcpg 49870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-13 2374  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fv 6497  df-1st 7930  df-2nd 7931  df-setrecs 49845  df-pg 49871

This theorem is referenced by:  pgind  49878