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Theorem fncnvimaeqv 16583
 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 6247 . 2 (𝐹 Fn V → (𝐹 “ V) = {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V})
2 fvex 6113 . . . . . 6 (𝐹𝑥) ∈ V
32a1i 11 . . . . 5 (𝐹 Fn V → (𝐹𝑥) ∈ V)
43biantrud 527 . . . 4 (𝐹 Fn V → (𝑥 ∈ V ↔ (𝑥 ∈ V ∧ (𝐹𝑥) ∈ V)))
5 fveq2 6103 . . . . . 6 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
65eleq1d 2672 . . . . 5 (𝑦 = 𝑥 → ((𝐹𝑦) ∈ V ↔ (𝐹𝑥) ∈ V))
76elrab 3331 . . . 4 (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} ↔ (𝑥 ∈ V ∧ (𝐹𝑥) ∈ V))
84, 7syl6rbbr 278 . . 3 (𝐹 Fn V → (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} ↔ 𝑥 ∈ V))
98eqrdv 2608 . 2 (𝐹 Fn V → {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} = V)
101, 9eqtrd 2644 1 (𝐹 Fn V → (𝐹 “ V) = V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ◡ccnv 5037   “ cima 5041   Fn wfn 5799  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812 This theorem is referenced by:  bascnvimaeqv  16584
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