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Theorem fnafv2elrn 43786
Description: An alternate function value belongs to the range of the function, analogous to fnfvelrn 6829. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
fnafv2elrn ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹''''𝐵) ∈ ran 𝐹)

Proof of Theorem fnafv2elrn
StepHypRef Expression
1 afv2elrn 43784 . 2 ((Fun 𝐹𝐵 ∈ dom 𝐹) → (𝐹''''𝐵) ∈ ran 𝐹)
21funfni 6432 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹''''𝐵) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2112  ran crn 5524   Fn wfn 6323  ''''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-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  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-nfc 2941  df-nel 3095  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-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6287  df-fun 6330  df-fn 6331  df-dfat 43672  df-afv2 43762
This theorem is referenced by:  fafv2elrn  43787  fafv2elrnb  43788
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