bj-elid4 - Mathbox for BJ

	Mathbox for BJ		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elid4		Structured version Visualization version GIF version

Theorem bj-elid4 37152

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 7963	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		eleq1 2816	. . . 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 6835	. . . . 5 ⊢ (2^nd ‘𝐴) ∈ V
5	4	inex2 5257	. . . 4 ⊢ ((1^st ‘𝐴) ∩ (2^nd ‘𝐴)) ∈ V
6		bj-opelid 37140	. . . 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 3436 ∩ cin 3902 ⟨cop 4583 I cid 5513 × cxp 5617 ‘cfv 6482 1^st c1st 7922 2^nd c2nd 7923

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 2701 ax-sep 5235 ax-nul 5245 ax-pr 5371 ax-un 7671

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-iota 6438 df-fun 6484 df-fv 6490 df-1st 7924 df-2nd 7925

This theorem is referenced by: bj-elid5 37153 bj-elid6 37154 bj-eldiag 37160

W3C validator