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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 6504 | . 2 ⊢ (𝐹 Fn V → (◡𝐹 “ V) = {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V}) | |
2 | fvexd 6366 | . . . . 5 ⊢ (𝐹 Fn V → (𝐹‘𝑥) ∈ V) | |
3 | 2 | biantrud 529 | . . . 4 ⊢ (𝐹 Fn V → (𝑥 ∈ V ↔ (𝑥 ∈ V ∧ (𝐹‘𝑥) ∈ V))) |
4 | fveq2 6354 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) | |
5 | 4 | eleq1d 2825 | . . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) ∈ V ↔ (𝐹‘𝑥) ∈ V)) |
6 | 5 | elrab 3505 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} ↔ (𝑥 ∈ V ∧ (𝐹‘𝑥) ∈ V)) |
7 | 3, 6 | syl6rbbr 279 | . . 3 ⊢ (𝐹 Fn V → (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} ↔ 𝑥 ∈ V)) |
8 | 7 | eqrdv 2759 | . 2 ⊢ (𝐹 Fn V → {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} = V) |
9 | 1, 8 | eqtrd 2795 | 1 ⊢ (𝐹 Fn V → (◡𝐹 “ V) = V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2140 {crab 3055 Vcvv 3341 ◡ccnv 5266 “ cima 5270 Fn wfn 6045 ‘cfv 6050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-sep 4934 ax-nul 4942 ax-pr 5056 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3343 df-sbc 3578 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-nul 4060 df-if 4232 df-sn 4323 df-pr 4325 df-op 4329 df-uni 4590 df-br 4806 df-opab 4866 df-id 5175 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-iota 6013 df-fun 6052 df-fn 6053 df-fv 6058 |
This theorem is referenced by: bascnvimaeqv 16983 |
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