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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 6189 | . . . 4 ⊢ ◡◡𝐵 = (𝐵 ∩ (V × V)) | |
2 | incom 4201 | . . . 4 ⊢ (𝐵 ∩ (V × V)) = ((V × V) ∩ 𝐵) | |
3 | 1, 2 | eqtri 2761 | . . 3 ⊢ ◡◡𝐵 = ((V × V) ∩ 𝐵) |
4 | 3 | eleq2i 2826 | . 2 ⊢ (𝐴 ∈ ◡◡𝐵 ↔ 𝐴 ∈ ((V × V) ∩ 𝐵)) |
5 | elinlem 42335 | . 2 ⊢ (𝐴 ∈ ((V × V) ∩ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵)) | |
6 | 4, 5 | bitri 275 | 1 ⊢ (𝐴 ∈ ◡◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∈ wcel 2107 Vcvv 3475 ∩ cin 3947 I cid 5573 × cxp 5674 ◡ccnv 5675 ‘cfv 6541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6493 df-fun 6543 df-fv 6549 |
This theorem is referenced by: (None) |
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