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Theorem focdmex 13716
Description: The codomain of an onto function is a set if its domain is a set. (Contributed by AV, 4-May-2021.)
Assertion
Ref Expression
focdmex ((𝐴𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)

Proof of Theorem focdmex
StepHypRef Expression
1 fof 6581 . . . . 5 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
21anim2i 619 . . . 4 ((𝐴𝑉𝐹:𝐴onto𝐵) → (𝐴𝑉𝐹:𝐴𝐵))
32ancomd 465 . . 3 ((𝐴𝑉𝐹:𝐴onto𝐵) → (𝐹:𝐴𝐵𝐴𝑉))
4 fex 6980 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
5 rnexg 7609 . . 3 (𝐹 ∈ V → ran 𝐹 ∈ V)
63, 4, 53syl 18 . 2 ((𝐴𝑉𝐹:𝐴onto𝐵) → ran 𝐹 ∈ V)
7 forn 6584 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87eleq1d 2900 . . 3 (𝐹:𝐴onto𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
98adantl 485 . 2 ((𝐴𝑉𝐹:𝐴onto𝐵) → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
106, 9mpbid 235 1 ((𝐴𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  Vcvv 3480  ran crn 5543  wf 6339  ontowfo 6341
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-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
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-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  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 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351
This theorem is referenced by:  hasheqf1oi  13717
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