Theorem elpglem3 49874

Description: Lemma for elpg 49875. (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 3441	. . . . . . . 8 ⊢ 𝑥 ∈ V
2		pweq 4565	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → 𝒫 𝑦 = 𝒫 𝑥)
3	2	sqxpeqd 5653	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝒫 𝑦 × 𝒫 𝑦) = (𝒫 𝑥 × 𝒫 𝑥))
4		eqid 2733	. . . . . . . . 9 ⊢ (𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦)) = (𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))
5	1	pwex 5322	. . . . . . . . . 10 ⊢ 𝒫 𝑥 ∈ V
6	5, 5	xpex 7695	. . . . . . . . 9 ⊢ (𝒫 𝑥 × 𝒫 𝑥) ∈ V
7	3, 4, 6	fvmpt 6938	. . . . . . . 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 2825	. . . . . 6 ⊢ (𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥) ↔ 𝐴 ∈ (𝒫 𝑥 × 𝒫 𝑥))
10		elxp7 7965	. . . . . 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 623	. . . 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 645	. . . 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 1849	. 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 1954	. 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 1541  ∃wex 1780   ∈ wcel 2113  Vcvv 3437   ⊆ wss 3898  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-ext 2705  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-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-iota 6445  df-fun 6491  df-fv 6497  df-1st 7930  df-2nd 7931

This theorem is referenced by:  elpg  49875