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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 7728 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | eleq1 2900 | . . . 4 ⊢ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 → (𝐴 ∈ I ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ I )) | |
3 | 2 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) → (𝐴 ∈ I ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ I )) |
4 | fvex 6683 | . . . . 5 ⊢ (2nd ‘𝐴) ∈ V | |
5 | 4 | inex2 5222 | . . . 4 ⊢ ((1st ‘𝐴) ∩ (2nd ‘𝐴)) ∈ V |
6 | bj-opelid 34451 | . . . 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 281 | . 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 〈cop 4573 I cid 5459 × cxp 5553 ‘cfv 6355 1st c1st 7687 2nd c2nd 7688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: bj-elid5 34464 bj-elid6 34465 bj-eldiag 34471 |
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