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Mirrors > Home > MPE Home > Th. List > fncnvimaeqv | Structured version Visualization version GIF version |
Description: The inverse images of the universal class V under functions on the universal class V are the universal class V itself. (Proposed by Mario Carneiro, 7-Mar-2020.) (Contributed by AV, 7-Mar-2020.) |
Ref | Expression |
---|---|
fncnvimaeqv | ⊢ (𝐹 Fn V → (◡𝐹 “ V) = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fncnvima2 6603 | . 2 ⊢ (𝐹 Fn V → (◡𝐹 “ V) = {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V}) | |
2 | fvexd 6461 | . . . . 5 ⊢ (𝐹 Fn V → (𝐹‘𝑥) ∈ V) | |
3 | 2 | biantrud 527 | . . . 4 ⊢ (𝐹 Fn V → (𝑥 ∈ V ↔ (𝑥 ∈ V ∧ (𝐹‘𝑥) ∈ V))) |
4 | fveq2 6446 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) | |
5 | 4 | eleq1d 2843 | . . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) ∈ V ↔ (𝐹‘𝑥) ∈ V)) |
6 | 5 | elrab 3571 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} ↔ (𝑥 ∈ V ∧ (𝐹‘𝑥) ∈ V)) |
7 | 3, 6 | syl6rbbr 282 | . . 3 ⊢ (𝐹 Fn V → (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} ↔ 𝑥 ∈ V)) |
8 | 7 | eqrdv 2775 | . 2 ⊢ (𝐹 Fn V → {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} = V) |
9 | 1, 8 | eqtrd 2813 | 1 ⊢ (𝐹 Fn V → (◡𝐹 “ V) = V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 {crab 3093 Vcvv 3397 ◡ccnv 5354 “ cima 5358 Fn wfn 6130 ‘cfv 6135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-fv 6143 |
This theorem is referenced by: bascnvimaeqv 17146 |
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