bj-elid4 - Mathbox for BJ

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Theorem bj-elid4 37342

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 7972	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		eleq1 2823	. . . 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 6846	. . . . 5 ⊢ (2^nd ‘𝐴) ∈ V
5	4	inex2 5262	. . . 4 ⊢ ((1^st ‘𝐴) ∩ (2^nd ‘𝐴)) ∈ V
6		bj-opelid 37330	. . . 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 688	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 1542 ∈ wcel 2114 Vcvv 3439 ∩ cin 3899 ⟨cop 4585 I cid 5517 × cxp 5621 ‘cfv 6491 1^st c1st 7931 2^nd c2nd 7932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-sep 5240 ax-nul 5250 ax-pr 5376 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rab 3399 df-v 3441 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4285 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-iota 6447 df-fun 6493 df-fv 6499 df-1st 7933 df-2nd 7934

This theorem is referenced by: bj-elid5 37343 bj-elid6 37344 bj-eldiag 37350

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