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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 5743 | . 2 ⊢ (𝐴 ∈ ◡𝐵 ↔ ∃𝑢∃𝑣(𝐴 = 〈𝑢, 𝑣〉 ∧ 〈𝑣, 𝑢〉 ∈ 𝐵)) | |
2 | fveq2 6665 | . . . . 5 ⊢ (𝐴 = 〈𝑢, 𝑣〉 → (𝐹‘𝐴) = (𝐹‘〈𝑢, 𝑣〉)) | |
3 | vex 3498 | . . . . . . 7 ⊢ 𝑢 ∈ V | |
4 | vex 3498 | . . . . . . 7 ⊢ 𝑣 ∈ V | |
5 | 3, 4 | opelvv 5589 | . . . . . 6 ⊢ 〈𝑢, 𝑣〉 ∈ (V × V) |
6 | 3, 4 | op2ndd 7694 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑢, 𝑣〉 → (2nd ‘𝑥) = 𝑣) |
7 | 3, 4 | op1std 7693 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑢, 𝑣〉 → (1st ‘𝑥) = 𝑢) |
8 | 6, 7 | opeq12d 4805 | . . . . . . 7 ⊢ (𝑥 = 〈𝑢, 𝑣〉 → 〈(2nd ‘𝑥), (1st ‘𝑥)〉 = 〈𝑣, 𝑢〉) |
9 | elcnvlem.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ (V × V) ↦ 〈(2nd ‘𝑥), (1st ‘𝑥)〉) | |
10 | opex 5349 | . . . . . . 7 ⊢ 〈𝑣, 𝑢〉 ∈ V | |
11 | 8, 9, 10 | fvmpt 6763 | . . . . . 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 2872 | . . . 4 ⊢ (𝐴 = 〈𝑢, 𝑣〉 → (𝐹‘𝐴) = 〈𝑣, 𝑢〉) |
14 | 13 | eleq1d 2897 | . . 3 ⊢ (𝐴 = 〈𝑢, 𝑣〉 → ((𝐹‘𝐴) ∈ 𝐵 ↔ 〈𝑣, 𝑢〉 ∈ 𝐵)) |
15 | 14 | copsex2gb 5674 | . 2 ⊢ (∃𝑢∃𝑣(𝐴 = 〈𝑢, 𝑣〉 ∧ 〈𝑣, 𝑢〉 ∈ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵)) |
16 | 1, 15 | bitri 277 | 1 ⊢ (𝐴 ∈ ◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3495 〈cop 4567 ↦ cmpt 5139 × cxp 5548 ◡ccnv 5549 ‘cfv 6350 1st c1st 7681 2nd c2nd 7682 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fv 6358 df-1st 7683 df-2nd 7684 |
This theorem is referenced by: elcnvintab 39955 |
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