bj-elid4 - Mathbox for BJ

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Theorem bj-elid4 37907

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 8028	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		eleq1 2850	. . . 4 ⊢ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ → (𝐴 ∈ I ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ I ))
3	2	adantl 487	. . 3 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩) → (𝐴 ∈ I ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ I ))
4		fvex 6895	. . . . 5 ⊢ (2^nd ‘𝐴) ∈ V
5	4	inex2 5285	. . . 4 ⊢ ((1^st ‘𝐴) ∩ (2^nd ‘𝐴)) ∈ V
6		bj-opelid 37895	. . . 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 14	. . 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 282	. 2 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩) → (𝐴 ∈ I ↔ (1^st ‘𝐴) = (2^nd ‘𝐴)))
9	1, 8	mpdan 700	1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1^st ‘𝐴) = (2^nd ‘𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3453 ∩ cin 3901 ⟨cop 4593 I cid 5553 × cxp 5657 ‘cfv 6537 1^st c1st 7987 2^nd c2nd 7988

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402 ax-un 7739

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-iota 6493 df-fun 6539 df-fv 6545 df-1st 7989 df-2nd 7990

This theorem is used by: bj-elid5 37908 bj-elid6 37909 bj-eldiag 37915

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