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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 7056 | . 2 ⊢ (𝐹 Fn V → (◡𝐹 “ V) = {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V}) | |
2 | fveq2 6885 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) | |
3 | 2 | eleq1d 2812 | . . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) ∈ V ↔ (𝐹‘𝑥) ∈ V)) |
4 | 3 | elrab 3678 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} ↔ (𝑥 ∈ V ∧ (𝐹‘𝑥) ∈ V)) |
5 | fvexd 6900 | . . . . 5 ⊢ (𝐹 Fn V → (𝐹‘𝑥) ∈ V) | |
6 | 5 | biantrud 531 | . . . 4 ⊢ (𝐹 Fn V → (𝑥 ∈ V ↔ (𝑥 ∈ V ∧ (𝐹‘𝑥) ∈ V))) |
7 | 4, 6 | bitr4id 290 | . . 3 ⊢ (𝐹 Fn V → (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} ↔ 𝑥 ∈ V)) |
8 | 7 | eqrdv 2724 | . 2 ⊢ (𝐹 Fn V → {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} = V) |
9 | 1, 8 | eqtrd 2766 | 1 ⊢ (𝐹 Fn V → (◡𝐹 “ V) = V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3426 Vcvv 3468 ◡ccnv 5668 “ cima 5672 Fn wfn 6532 ‘cfv 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-fv 6545 |
This theorem is referenced by: bascnvimaeqv 18084 |
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