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Theorem fcdmvafv2v 47186
Description: If the codomain of a function is a set, the alternate function value is always also a set. (Contributed by AV, 4-Sep-2022.)
Assertion
Ref Expression
fcdmvafv2v ((𝐹:𝐴𝐵𝐵𝑉) → (𝐹''''𝐶) ∈ V)

Proof of Theorem fcdmvafv2v
StepHypRef Expression
1 df-f 6567 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 ssexg 5329 . . . . 5 ((ran 𝐹𝐵𝐵𝑉) → ran 𝐹 ∈ V)
32ex 412 . . . 4 (ran 𝐹𝐵 → (𝐵𝑉 → ran 𝐹 ∈ V))
41, 3simplbiim 504 . . 3 (𝐹:𝐴𝐵 → (𝐵𝑉 → ran 𝐹 ∈ V))
54imp 406 . 2 ((𝐹:𝐴𝐵𝐵𝑉) → ran 𝐹 ∈ V)
6 afv2ex 47164 . 2 (ran 𝐹 ∈ V → (𝐹''''𝐶) ∈ V)
75, 6syl 17 1 ((𝐹:𝐴𝐵𝐵𝑉) → (𝐹''''𝐶) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  Vcvv 3478  wss 3963  ran crn 5690   Fn wfn 6558  wf 6559  ''''cafv2 47158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516  df-f 6567  df-afv2 47159
This theorem is referenced by: (None)
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