Theorem elpglem3 48811

Description: Lemma for elpg 48812. (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 3492	. . . . . . . 8 ⊢ 𝑥 ∈ V
2		pweq 4636	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → 𝒫 𝑦 = 𝒫 𝑥)
3	2	sqxpeqd 5732	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝒫 𝑦 × 𝒫 𝑦) = (𝒫 𝑥 × 𝒫 𝑥))
4		eqid 2740	. . . . . . . . 9 ⊢ (𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦)) = (𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))
5	1	pwex 5398	. . . . . . . . . 10 ⊢ 𝒫 𝑥 ∈ V
6	5, 5	xpex 7790	. . . . . . . . 9 ⊢ (𝒫 𝑥 × 𝒫 𝑥) ∈ V
7	3, 4, 6	fvmpt 7031	. . . . . . . 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 2836	. . . . . 6 ⊢ (𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥) ↔ 𝐴 ∈ (𝒫 𝑥 × 𝒫 𝑥))
10		elxp7 8067	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 𝑥 × 𝒫 𝑥) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))
11	9, 10	bitri 275	. . . . 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 644	. . . 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 275	. . 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 1846	. 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 1953	. 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 275	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 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622   ↦ cmpt 5249   × cxp 5698  ‘cfv 6575  1^st c1st 8030  2^nd c2nd 8031  Pgcpg 48807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6527  df-fun 6577  df-fv 6583  df-1st 8032  df-2nd 8033

This theorem is referenced by:  elpg  48812