Theorem elpglem3 46304

Description: Lemma for elpg 46305. (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 3426	. . . . . . . 8 ⊢ 𝑥 ∈ V
2		pweq 4546	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → 𝒫 𝑦 = 𝒫 𝑥)
3	2	sqxpeqd 5612	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝒫 𝑦 × 𝒫 𝑦) = (𝒫 𝑥 × 𝒫 𝑥))
4		eqid 2738	. . . . . . . . 9 ⊢ (𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦)) = (𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))
5	1	pwex 5298	. . . . . . . . . 10 ⊢ 𝒫 𝑥 ∈ V
6	5, 5	xpex 7581	. . . . . . . . 9 ⊢ (𝒫 𝑥 × 𝒫 𝑥) ∈ V
7	3, 4, 6	fvmpt 6857	. . . . . . . 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 2830	. . . . . 6 ⊢ (𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥) ↔ 𝐴 ∈ (𝒫 𝑥 × 𝒫 𝑥))
10		elxp7 7839	. . . . . 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 622	. . . 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 641	. . . 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 1851	. 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 1958	. 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 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530   ↦ cmpt 5153   × cxp 5578  ‘cfv 6418  1^st c1st 7802  2^nd c2nd 7803  Pgcpg 46300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-1st 7804  df-2nd 7805

This theorem is referenced by:  elpg  46305