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Theorem fncnvimaeqv 16961
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 6502 . 2 (𝐹 Fn V → (𝐹 “ V) = {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V})
2 fvexd 6364 . . . . 5 (𝐹 Fn V → (𝐹𝑥) ∈ V)
32biantrud 529 . . . 4 (𝐹 Fn V → (𝑥 ∈ V ↔ (𝑥 ∈ V ∧ (𝐹𝑥) ∈ V)))
4 fveq2 6352 . . . . . 6 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
54eleq1d 2824 . . . . 5 (𝑦 = 𝑥 → ((𝐹𝑦) ∈ V ↔ (𝐹𝑥) ∈ V))
65elrab 3504 . . . 4 (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} ↔ (𝑥 ∈ V ∧ (𝐹𝑥) ∈ V))
73, 6syl6rbbr 279 . . 3 (𝐹 Fn V → (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} ↔ 𝑥 ∈ V))
87eqrdv 2758 . 2 (𝐹 Fn V → {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} = V)
91, 8eqtrd 2794 1 (𝐹 Fn V → (𝐹 “ V) = V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  ccnv 5265  cima 5269   Fn wfn 6044  cfv 6049
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 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
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 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  bascnvimaeqv  16962
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