MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fncnvimaeqv Structured version   Visualization version   GIF version

Theorem fncnvimaeqv 17368
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 6820 . 2 (𝐹 Fn V → (𝐹 “ V) = {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V})
2 fvexd 6674 . . . . 5 (𝐹 Fn V → (𝐹𝑥) ∈ V)
32biantrud 535 . . . 4 (𝐹 Fn V → (𝑥 ∈ V ↔ (𝑥 ∈ V ∧ (𝐹𝑥) ∈ V)))
4 fveq2 6659 . . . . . 6 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
54eleq1d 2900 . . . . 5 (𝑦 = 𝑥 → ((𝐹𝑦) ∈ V ↔ (𝐹𝑥) ∈ V))
65elrab 3666 . . . 4 (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} ↔ (𝑥 ∈ V ∧ (𝐹𝑥) ∈ V))
73, 6syl6rbbr 293 . . 3 (𝐹 Fn V → (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} ↔ 𝑥 ∈ V))
87eqrdv 2822 . 2 (𝐹 Fn V → {𝑦 ∈ V ∣ (𝐹𝑦) ∈ V} = V)
91, 8eqtrd 2859 1 (𝐹 Fn V → (𝐹 “ V) = V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {crab 3137  Vcvv 3480  ccnv 5542  cima 5546   Fn wfn 6339  cfv 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-fv 6352
This theorem is referenced by:  bascnvimaeqv  17369
  Copyright terms: Public domain W3C validator