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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 8008 | . 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 481 | . . 3 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) → (𝐴 ∈ I ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ I )) |
4 | fvex 6895 | . . . . 5 ⊢ (2nd ‘𝐴) ∈ V | |
5 | 4 | inex2 5309 | . . . 4 ⊢ ((1st ‘𝐴) ∩ (2nd ‘𝐴)) ∈ V |
6 | bj-opelid 36528 | . . . 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 279 | . 2 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴))) |
9 | 1, 8 | mpdan 684 | 1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∩ cin 3940 〈cop 4627 I cid 5564 × cxp 5665 ‘cfv 6534 1st c1st 7967 2nd c2nd 7968 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6486 df-fun 6536 df-fv 6542 df-1st 7969 df-2nd 7970 |
This theorem is referenced by: bj-elid5 36541 bj-elid6 36542 bj-eldiag 36548 |
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