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Theorem fncnvimaeqv 17145
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.)
Assertion
Ref Expression
fncnvimaeqv (𝐹 Fn V → (𝐹 “ V) = V)

Proof of Theorem fncnvimaeqv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fncnvima2 6603 . 2 (𝐹 Fn V → (𝐹 “ V) = {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V})
2 fvexd 6461 . . . . 5 (𝐹 Fn V → (𝐹𝑥) ∈ V)
32biantrud 527 . . . 4 (𝐹 Fn V → (𝑥 ∈ V ↔ (𝑥 ∈ V ∧ (𝐹𝑥) ∈ V)))
4 fveq2 6446 . . . . . 6 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
54eleq1d 2843 . . . . 5 (𝑦 = 𝑥 → ((𝐹𝑦) ∈ V ↔ (𝐹𝑥) ∈ V))
65elrab 3571 . . . 4 (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} ↔ (𝑥 ∈ V ∧ (𝐹𝑥) ∈ V))
73, 6syl6rbbr 282 . . 3 (𝐹 Fn V → (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} ↔ 𝑥 ∈ V))
87eqrdv 2775 . 2 (𝐹 Fn V → {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} = V)
91, 8eqtrd 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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