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Theorem nfafv2 44954
Description: Bound-variable hypothesis builder for function value, analogous to nffv 6814. To prove a deduction version of this analogous to nffvd 6816 is not easily possible because a deduction version of nfdfat 44863 cannot be shown easily. (Contributed by AV, 4-Sep-2022.)
Hypotheses
Ref Expression
nfafv2.1 𝑥𝐹
nfafv2.2 𝑥𝐴
Assertion
Ref Expression
nfafv2 𝑥(𝐹''''𝐴)

Proof of Theorem nfafv2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-afv2 44945 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ran 𝐹)
2 nfafv2.1 . . . 4 𝑥𝐹
3 nfafv2.2 . . . 4 𝑥𝐴
42, 3nfdfat 44863 . . 3 𝑥 𝐹 defAt 𝐴
5 nfcv 2905 . . . . 5 𝑥𝑦
63, 2, 5nfbr 5128 . . . 4 𝑥 𝐴𝐹𝑦
76nfiotaw 6414 . . 3 𝑥(℩𝑦𝐴𝐹𝑦)
82nfrn 5873 . . . . 5 𝑥ran 𝐹
98nfuni 4851 . . . 4 𝑥 ran 𝐹
109nfpw 4558 . . 3 𝑥𝒫 ran 𝐹
114, 7, 10nfif 4495 . 2 𝑥if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ran 𝐹)
121, 11nfcxfr 2903 1 𝑥(𝐹''''𝐴)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2885  ifcif 4465  𝒫 cpw 4539   cuni 4844   class class class wbr 5081  ran crn 5601  cio 6408   defAt wdfat 44852  ''''cafv2 44944
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-iota 6410  df-fun 6460  df-dfat 44855  df-afv2 44945
This theorem is referenced by:  csbafv212g  44955
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