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Theorem focdmex 13456
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 6366 . . . . 5 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
21anim2i 610 . . . 4 ((𝐴𝑉𝐹:𝐴onto𝐵) → (𝐴𝑉𝐹:𝐴𝐵))
32ancomd 455 . . 3 ((𝐴𝑉𝐹:𝐴onto𝐵) → (𝐹:𝐴𝐵𝐴𝑉))
4 fex 6761 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
5 rnexg 7376 . . 3 (𝐹 ∈ V → ran 𝐹 ∈ V)
63, 4, 53syl 18 . 2 ((𝐴𝑉𝐹:𝐴onto𝐵) → ran 𝐹 ∈ V)
7 forn 6369 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87eleq1d 2844 . . 3 (𝐹:𝐴onto𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
98adantl 475 . 2 ((𝐴𝑉𝐹:𝐴onto𝐵) → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
106, 9mpbid 224 1 ((𝐴𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wcel 2107  Vcvv 3398  ran crn 5356  wf 6131  ontowfo 6133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143
This theorem is referenced by:  hasheqf1oi  13457
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