Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frnvafv2v Structured version   Visualization version   GIF version

Theorem frnvafv2v 43789
 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
frnvafv2v ((𝐹:𝐴𝐵𝐵𝑉) → (𝐹''''𝐶) ∈ V)

Proof of Theorem frnvafv2v
StepHypRef Expression
1 df-f 6332 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 ssexg 5194 . . . . 5 ((ran 𝐹𝐵𝐵𝑉) → ran 𝐹 ∈ V)
32ex 416 . . . 4 (ran 𝐹𝐵 → (𝐵𝑉 → ran 𝐹 ∈ V))
41, 3simplbiim 508 . . 3 (𝐹:𝐴𝐵 → (𝐵𝑉 → ran 𝐹 ∈ V))
54imp 410 . 2 ((𝐹:𝐴𝐵𝐵𝑉) → ran 𝐹 ∈ V)
6 afv2ex 43767 . 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 399   ∈ wcel 2112  Vcvv 3444   ⊆ wss 3884  ran crn 5524   Fn wfn 6323  ⟶wf 6324  ''''cafv2 43761 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-uni 4804  df-iota 6287  df-f 6332  df-afv2 43762 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator