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Theorem fcdmvafv2v 47248
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 6565 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 ssexg 5323 . . . . 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 47226 . 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 2108  Vcvv 3480  wss 3951  ran crn 5686   Fn wfn 6556  wf 6557  ''''cafv2 47220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-uni 4908  df-iota 6514  df-f 6565  df-afv2 47221
This theorem is referenced by: (None)
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