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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 5547 | . 2 ⊢ (𝐴 ∈ ◡𝐵 ↔ ∃𝑢∃𝑣(𝐴 = 〈𝑢, 𝑣〉 ∧ 〈𝑣, 𝑢〉 ∈ 𝐵)) | |
2 | fveq2 6448 | . . . . 5 ⊢ (𝐴 = 〈𝑢, 𝑣〉 → (𝐹‘𝐴) = (𝐹‘〈𝑢, 𝑣〉)) | |
3 | vex 3401 | . . . . . . 7 ⊢ 𝑢 ∈ V | |
4 | vex 3401 | . . . . . . 7 ⊢ 𝑣 ∈ V | |
5 | 3, 4 | opelvv 5396 | . . . . . 6 ⊢ 〈𝑢, 𝑣〉 ∈ (V × V) |
6 | 3, 4 | op2ndd 7458 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑢, 𝑣〉 → (2nd ‘𝑥) = 𝑣) |
7 | 3, 4 | op1std 7457 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑢, 𝑣〉 → (1st ‘𝑥) = 𝑢) |
8 | 6, 7 | opeq12d 4646 | . . . . . . 7 ⊢ (𝑥 = 〈𝑢, 𝑣〉 → 〈(2nd ‘𝑥), (1st ‘𝑥)〉 = 〈𝑣, 𝑢〉) |
9 | elcnvlem.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ (V × V) ↦ 〈(2nd ‘𝑥), (1st ‘𝑥)〉) | |
10 | opex 5166 | . . . . . . 7 ⊢ 〈𝑣, 𝑢〉 ∈ V | |
11 | 8, 9, 10 | fvmpt 6544 | . . . . . 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 | syl6eq 2830 | . . . 4 ⊢ (𝐴 = 〈𝑢, 𝑣〉 → (𝐹‘𝐴) = 〈𝑣, 𝑢〉) |
14 | 13 | eleq1d 2844 | . . 3 ⊢ (𝐴 = 〈𝑢, 𝑣〉 → ((𝐹‘𝐴) ∈ 𝐵 ↔ 〈𝑣, 𝑢〉 ∈ 𝐵)) |
15 | 14 | copsex2gb 5479 | . 2 ⊢ (∃𝑢∃𝑣(𝐴 = 〈𝑢, 𝑣〉 ∧ 〈𝑣, 𝑢〉 ∈ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵)) |
16 | 1, 15 | bitri 267 | 1 ⊢ (𝐴 ∈ ◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1601 ∃wex 1823 ∈ wcel 2107 Vcvv 3398 〈cop 4404 ↦ cmpt 4967 × cxp 5355 ◡ccnv 5356 ‘cfv 6137 1st c1st 7445 2nd c2nd 7446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-iota 6101 df-fun 6139 df-fv 6145 df-1st 7447 df-2nd 7448 |
This theorem is referenced by: elcnvintab 38879 |
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