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Theorem bj-elid4 35266

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 7843	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		eleq1 2826	. . . 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 6769	. . . . 5 ⊢ (2^nd ‘𝐴) ∈ V
5	4	inex2 5237	. . . 4 ⊢ ((1^st ‘𝐴) ∩ (2^nd ‘𝐴)) ∈ V
6		bj-opelid 35254	. . . 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 278	. 2 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩) → (𝐴 ∈ I ↔ (1^st ‘𝐴) = (2^nd ‘𝐴)))
9	1, 8	mpdan 683	1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1^st ‘𝐴) = (2^nd ‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ∩ cin 3882 ⟨cop 4564 I cid 5479 × cxp 5578 ‘cfv 6418 1^st c1st 7802 2^nd c2nd 7803

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-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-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: bj-elid5 35267 bj-elid6 35268 bj-eldiag 35274

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