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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elcnvcnvlem | Structured version Visualization version GIF version |
Description: Two ways to say a set is a member of the converse of the converse of a class. (Contributed by RP, 20-Aug-2020.) |
Ref | Expression |
---|---|
elcnvcnvlem | ⊢ (𝐴 ∈ ◡◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnv 5889 | . . . 4 ⊢ ◡◡𝐵 = (𝐵 ∩ (V × V)) | |
2 | incom 4066 | . . . 4 ⊢ (𝐵 ∩ (V × V)) = ((V × V) ∩ 𝐵) | |
3 | 1, 2 | eqtri 2802 | . . 3 ⊢ ◡◡𝐵 = ((V × V) ∩ 𝐵) |
4 | 3 | eleq2i 2857 | . 2 ⊢ (𝐴 ∈ ◡◡𝐵 ↔ 𝐴 ∈ ((V × V) ∩ 𝐵)) |
5 | elinlem 39326 | . 2 ⊢ (𝐴 ∈ ((V × V) ∩ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵)) | |
6 | 4, 5 | bitri 267 | 1 ⊢ (𝐴 ∈ ◡◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 ∈ wcel 2050 Vcvv 3415 ∩ cin 3828 I cid 5311 × cxp 5405 ◡ccnv 5406 ‘cfv 6188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-iota 6152 df-fun 6190 df-fv 6196 |
This theorem is referenced by: (None) |
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