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Theorem fcdmvafv2v 47253
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 6564 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 ssexg 5322 . . . . 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 47231 . 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 2107  Vcvv 3479  wss 3950  ran crn 5685   Fn wfn 6555  wf 6556  ''''cafv2 47225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-uni 4907  df-iota 6513  df-f 6564  df-afv2 47226
This theorem is referenced by: (None)
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