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Theorem fncnvimaeqv 18165
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 7080 . 2 (𝐹 Fn V → (𝐹 “ V) = {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V})
2 fveq2 6905 . . . . . 6 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
32eleq1d 2825 . . . . 5 (𝑦 = 𝑥 → ((𝐹𝑦) ∈ V ↔ (𝐹𝑥) ∈ V))
43elrab 3691 . . . 4 (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} ↔ (𝑥 ∈ V ∧ (𝐹𝑥) ∈ V))
5 fvexd 6920 . . . . 5 (𝐹 Fn V → (𝐹𝑥) ∈ V)
65biantrud 531 . . . 4 (𝐹 Fn V → (𝑥 ∈ V ↔ (𝑥 ∈ V ∧ (𝐹𝑥) ∈ V)))
74, 6bitr4id 290 . . 3 (𝐹 Fn V → (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} ↔ 𝑥 ∈ V))
87eqrdv 2734 . 2 (𝐹 Fn V → {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} = V)
91, 8eqtrd 2776 1 (𝐹 Fn V → (𝐹 “ V) = V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {crab 3435  Vcvv 3479  ccnv 5683  cima 5687   Fn wfn 6555  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568
This theorem is referenced by:  bascnvimaeqv  18166
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