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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 6055 | . . . 4 ⊢ ◡◡𝐵 = (𝐵 ∩ (V × V)) | |
2 | incom 4115 | . . . 4 ⊢ (𝐵 ∩ (V × V)) = ((V × V) ∩ 𝐵) | |
3 | 1, 2 | eqtri 2765 | . . 3 ⊢ ◡◡𝐵 = ((V × V) ∩ 𝐵) |
4 | 3 | eleq2i 2829 | . 2 ⊢ (𝐴 ∈ ◡◡𝐵 ↔ 𝐴 ∈ ((V × V) ∩ 𝐵)) |
5 | elinlem 40882 | . 2 ⊢ (𝐴 ∈ ((V × V) ∩ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵)) | |
6 | 4, 5 | bitri 278 | 1 ⊢ (𝐴 ∈ ◡◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2110 Vcvv 3408 ∩ cin 3865 I cid 5454 × cxp 5549 ◡ccnv 5550 ‘cfv 6380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 |
This theorem is referenced by: (None) |
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