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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 5300 | . 2 ⊢ (𝐴 ∈ ◡𝐵 ↔ ∃𝑢∃𝑣(𝐴 = 〈𝑢, 𝑣〉 ∧ 〈𝑣, 𝑢〉 ∈ 𝐵)) | |
2 | fveq2 6191 | . . . . 5 ⊢ (𝐴 = 〈𝑢, 𝑣〉 → (𝐹‘𝐴) = (𝐹‘〈𝑢, 𝑣〉)) | |
3 | vex 3203 | . . . . . . 7 ⊢ 𝑢 ∈ V | |
4 | vex 3203 | . . . . . . 7 ⊢ 𝑣 ∈ V | |
5 | 3, 4 | opelvv 5166 | . . . . . 6 ⊢ 〈𝑢, 𝑣〉 ∈ (V × V) |
6 | 3, 4 | op2ndd 7179 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑢, 𝑣〉 → (2nd ‘𝑥) = 𝑣) |
7 | 3, 4 | op1std 7178 | . . . . . . . 8 ⊢ (𝑥 = 〈𝑢, 𝑣〉 → (1st ‘𝑥) = 𝑢) |
8 | 6, 7 | opeq12d 4410 | . . . . . . 7 ⊢ (𝑥 = 〈𝑢, 𝑣〉 → 〈(2nd ‘𝑥), (1st ‘𝑥)〉 = 〈𝑣, 𝑢〉) |
9 | elcnvlem.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ (V × V) ↦ 〈(2nd ‘𝑥), (1st ‘𝑥)〉) | |
10 | opex 4932 | . . . . . . 7 ⊢ 〈𝑣, 𝑢〉 ∈ V | |
11 | 8, 9, 10 | fvmpt 6282 | . . . . . 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 2672 | . . . 4 ⊢ (𝐴 = 〈𝑢, 𝑣〉 → (𝐹‘𝐴) = 〈𝑣, 𝑢〉) |
14 | 13 | eleq1d 2686 | . . 3 ⊢ (𝐴 = 〈𝑢, 𝑣〉 → ((𝐹‘𝐴) ∈ 𝐵 ↔ 〈𝑣, 𝑢〉 ∈ 𝐵)) |
15 | 14 | copsex2gb 5230 | . 2 ⊢ (∃𝑢∃𝑣(𝐴 = 〈𝑢, 𝑣〉 ∧ 〈𝑣, 𝑢〉 ∈ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵)) |
16 | 1, 15 | bitri 264 | 1 ⊢ (𝐴 ∈ ◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 〈cop 4183 ↦ cmpt 4729 × cxp 5112 ◡ccnv 5113 ‘cfv 5888 1st c1st 7166 2nd c2nd 7167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: elcnvintab 37908 |
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