Theorem elpglem3 48330

Description: Lemma for elpg 48331. (Contributed by Emmett Weisz, 28-Aug-2021.)

Assertion

Ref	Expression
elpglem3	⊢ (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem elpglem3

Step	Hyp	Ref	Expression
1		vex 3465	. . . . . . . 8 ⊢ 𝑥 ∈ V
2		pweq 4618	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → 𝒫 𝑦 = 𝒫 𝑥)
3	2	sqxpeqd 5710	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝒫 𝑦 × 𝒫 𝑦) = (𝒫 𝑥 × 𝒫 𝑥))
4		eqid 2725	. . . . . . . . 9 ⊢ (𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦)) = (𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))
5	1	pwex 5380	. . . . . . . . . 10 ⊢ 𝒫 𝑥 ∈ V
6	5, 5	xpex 7756	. . . . . . . . 9 ⊢ (𝒫 𝑥 × 𝒫 𝑥) ∈ V
7	3, 4, 6	fvmpt 7004	. . . . . . . 8 ⊢ (𝑥 ∈ V → ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥) = (𝒫 𝑥 × 𝒫 𝑥))
8	1, 7	ax-mp 5	. . . . . . 7 ⊢ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥) = (𝒫 𝑥 × 𝒫 𝑥)
9	8	eleq2i 2817	. . . . . 6 ⊢ (𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥) ↔ 𝐴 ∈ (𝒫 𝑥 × 𝒫 𝑥))
10		elxp7 8029	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 𝑥 × 𝒫 𝑥) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))
11	9, 10	bitri 274	. . . . 5 ⊢ (𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))
12	11	anbi2i 621	. . . 4 ⊢ ((𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝑥 ⊆ Pg ∧ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))))
13		an12 643	. . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))) ↔ (𝐴 ∈ (V × V) ∧ (𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))))
14	12, 13	bitri 274	. . 3 ⊢ ((𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ (𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))))
15	14	exbii 1842	. 2 ⊢ (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ ∃𝑥(𝐴 ∈ (V × V) ∧ (𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))))
16		19.42v 1949	. 2 ⊢ (∃𝑥(𝐴 ∈ (V × V) ∧ (𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))))
17	15, 16	bitri 274	1 ⊢ (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3461   ⊆ wss 3944  𝒫 cpw 4604   ↦ cmpt 5232   × cxp 5676  ‘cfv 6549  1^st c1st 7992  2^nd c2nd 7993  Pgcpg 48326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fv 6557  df-1st 7994  df-2nd 7995

This theorem is referenced by:  elpg  48331