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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvepima | Structured version Visualization version GIF version |
Description: The image of converse epsilon. (Contributed by Peter Mazsa, 22-Mar-2023.) |
Ref | Expression |
---|---|
cnvepima | ⊢ (𝐴 ∈ 𝑉 → (◡ E “ 𝐴) = ∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qsid 8213 | . . 3 ⊢ (𝐴 / ◡ E ) = 𝐴 | |
2 | 1 | unieqi 4754 | . 2 ⊢ ∪ (𝐴 / ◡ E ) = ∪ 𝐴 |
3 | cnvepresex 35123 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V) | |
4 | uniqsALTV 35118 | . . 3 ⊢ ((◡ E ↾ 𝐴) ∈ V → ∪ (𝐴 / ◡ E ) = (◡ E “ 𝐴)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ (𝐴 / ◡ E ) = (◡ E “ 𝐴)) |
6 | 2, 5 | syl5reqr 2846 | 1 ⊢ (𝐴 ∈ 𝑉 → (◡ E “ 𝐴) = ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ∪ cuni 4745 E cep 5352 ◡ccnv 5442 ↾ cres 5445 “ cima 5446 / cqs 8138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-eprel 5353 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ec 8141 df-qs 8145 |
This theorem is referenced by: (None) |
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