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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elcnvlem | Structured version Visualization version GIF version |
Description: Two ways to say a set is a member of the converse of a class. (Contributed by RP, 19-Aug-2020.) |
Ref | Expression |
---|---|
elcnvlem.f | ⊢ 𝐹 = (𝑥 ∈ (V × V) ↦ 〈(2nd ‘𝑥), (1st ‘𝑥)〉) |
Ref | Expression |
---|---|
elcnvlem | ⊢ (𝐴 ∈ ◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elcnv2 5891 | . 2 ⊢ (𝐴 ∈ ◡𝐵 ↔ ∃𝑢∃𝑣(𝐴 = 〈𝑢, 𝑣〉 ∧ 〈𝑣, 𝑢〉 ∈ 𝐵)) | |
2 | fveq2 6907 | . . . . 5 ⊢ (𝐴 = 〈𝑢, 𝑣〉 → (𝐹‘𝐴) = (𝐹‘〈𝑢, 𝑣〉)) | |
3 | vex 3482 | . . . . . . 7 ⊢ 𝑢 ∈ V | |
4 | vex 3482 | . . . . . . 7 ⊢ 𝑣 ∈ V | |
5 | 3, 4 | opelvv 5729 | . . . . . 6 ⊢ 〈𝑢, 𝑣〉 ∈ (V × V) |
6 | 3, 4 | op2ndd 8024 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑢, 𝑣〉 → (2nd ‘𝑥) = 𝑣) |
7 | 3, 4 | op1std 8023 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑢, 𝑣〉 → (1st ‘𝑥) = 𝑢) |
8 | 6, 7 | opeq12d 4886 | . . . . . . 7 ⊢ (𝑥 = 〈𝑢, 𝑣〉 → 〈(2nd ‘𝑥), (1st ‘𝑥)〉 = 〈𝑣, 𝑢〉) |
9 | elcnvlem.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ (V × V) ↦ 〈(2nd ‘𝑥), (1st ‘𝑥)〉) | |
10 | opex 5475 | . . . . . . 7 ⊢ 〈𝑣, 𝑢〉 ∈ V | |
11 | 8, 9, 10 | fvmpt 7016 | . . . . . 6 ⊢ (〈𝑢, 𝑣〉 ∈ (V × V) → (𝐹‘〈𝑢, 𝑣〉) = 〈𝑣, 𝑢〉) |
12 | 5, 11 | ax-mp 5 | . . . . 5 ⊢ (𝐹‘〈𝑢, 𝑣〉) = 〈𝑣, 𝑢〉 |
13 | 2, 12 | eqtrdi 2791 | . . . 4 ⊢ (𝐴 = 〈𝑢, 𝑣〉 → (𝐹‘𝐴) = 〈𝑣, 𝑢〉) |
14 | 13 | eleq1d 2824 | . . 3 ⊢ (𝐴 = 〈𝑢, 𝑣〉 → ((𝐹‘𝐴) ∈ 𝐵 ↔ 〈𝑣, 𝑢〉 ∈ 𝐵)) |
15 | 14 | copsex2gb 5819 | . 2 ⊢ (∃𝑢∃𝑣(𝐴 = 〈𝑢, 𝑣〉 ∧ 〈𝑣, 𝑢〉 ∈ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵)) |
16 | 1, 15 | bitri 275 | 1 ⊢ (𝐴 ∈ ◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ↦ cmpt 5231 × cxp 5687 ◡ccnv 5688 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: elcnvintab 43592 |
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