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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvepimaex | Structured version Visualization version GIF version |
Description: The image of converse epsilon exists, proof via imaexALTV 35623 (see also cnvepima 35630 and uniexg 7463 for alternate way). (Contributed by Peter Mazsa, 22-Mar-2023.) |
Ref | Expression |
---|---|
cnvepimaex | ⊢ (𝐴 ∈ 𝑉 → (◡ E “ 𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvepresex 35627 | . 2 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V) | |
2 | imaexALTV 35623 | . . 3 ⊢ ((◡ E ∈ V ∨ ((◡ E ↾ 𝐴) ∈ V ∧ 𝐴 ∈ 𝑉)) → (◡ E “ 𝐴) ∈ V) | |
3 | 2 | olcs 872 | . 2 ⊢ (((◡ E ↾ 𝐴) ∈ V ∧ 𝐴 ∈ 𝑉) → (◡ E “ 𝐴) ∈ V) |
4 | 1, 3 | mpancom 686 | 1 ⊢ (𝐴 ∈ 𝑉 → (◡ E “ 𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 Vcvv 3493 E cep 5461 ◡ccnv 5551 ↾ cres 5554 “ cima 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-id 5457 df-eprel 5462 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-ec 8288 df-qs 8292 |
This theorem is referenced by: (None) |
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