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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elid4 | Structured version Visualization version GIF version |
Description: Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.) |
Ref | Expression |
---|---|
bj-elid4 | ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1st2nd2 8030 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩) | |
2 | eleq1 2813 | . . . 4 ⊢ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ → (𝐴 ∈ I ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I )) | |
3 | 2 | adantl 480 | . . 3 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩) → (𝐴 ∈ I ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I )) |
4 | fvex 6905 | . . . . 5 ⊢ (2nd ‘𝐴) ∈ V | |
5 | 4 | inex2 5313 | . . . 4 ⊢ ((1st ‘𝐴) ∩ (2nd ‘𝐴)) ∈ V |
6 | bj-opelid 36692 | . . . 4 ⊢ (((1st ‘𝐴) ∩ (2nd ‘𝐴)) ∈ V → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴))) | |
7 | 5, 6 | mp1i 13 | . . 3 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴))) |
8 | 3, 7 | bitrd 278 | . 2 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴))) |
9 | 1, 8 | mpdan 685 | 1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∩ cin 3938 ⟨cop 4630 I cid 5569 × cxp 5670 ‘cfv 6543 1st c1st 7989 2nd c2nd 7990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6495 df-fun 6545 df-fv 6551 df-1st 7991 df-2nd 7992 |
This theorem is referenced by: bj-elid5 36705 bj-elid6 36706 bj-eldiag 36712 |
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