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Theorem bj-elid4 37163

Description: Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)

**Assertion**
Ref	Expression
bj-elid4	⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1^st ‘𝐴) = (2^nd ‘𝐴)))

**Proof of Theorem bj-elid4**
Step	Hyp	Ref	Expression
1		1st2nd2 8010	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		eleq1 2817	. . . 4 ⊢ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ → (𝐴 ∈ I ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ I ))
3	2	adantl 481	. . 3 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩) → (𝐴 ∈ I ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ I ))
4		fvex 6874	. . . . 5 ⊢ (2^nd ‘𝐴) ∈ V
5	4	inex2 5276	. . . 4 ⊢ ((1^st ‘𝐴) ∩ (2^nd ‘𝐴)) ∈ V
6		bj-opelid 37151	. . . 4 ⊢ (((1^st ‘𝐴) ∩ (2^nd ‘𝐴)) ∈ V → (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ I ↔ (1^st ‘𝐴) = (2^nd ‘𝐴)))
7	5, 6	mp1i 13	. . 3 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩) → (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ I ↔ (1^st ‘𝐴) = (2^nd ‘𝐴)))
8	3, 7	bitrd 279	. 2 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩) → (𝐴 ∈ I ↔ (1^st ‘𝐴) = (2^nd ‘𝐴)))
9	1, 8	mpdan 687	1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1^st ‘𝐴) = (2^nd ‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3450 ∩ cin 3916 ⟨cop 4598 I cid 5535 × cxp 5639 ‘cfv 6514 1^st c1st 7969 2^nd c2nd 7970

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-iota 6467 df-fun 6516 df-fv 6522 df-1st 7971 df-2nd 7972

This theorem is referenced by: bj-elid5 37164 bj-elid6 37165 bj-eldiag 37171

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