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Theorem nfafv2 41898
Description: Bound-variable hypothesis builder for function value, analogous to nffv 6385. To prove a deduction version of this analogous to nffvd 6387 is not easily possible because a deduction version of nfdfat 41807 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 41889 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ran 𝐹)
2 nfafv2.1 . . . 4 𝑥𝐹
3 nfafv2.2 . . . 4 𝑥𝐴
42, 3nfdfat 41807 . . 3 𝑥 𝐹 defAt 𝐴
5 nfcv 2907 . . . . 5 𝑥𝑦
63, 2, 5nfbr 4856 . . . 4 𝑥 𝐴𝐹𝑦
76nfiota 6035 . . 3 𝑥(℩𝑦𝐴𝐹𝑦)
82nfrn 5537 . . . . 5 𝑥ran 𝐹
98nfuni 4600 . . . 4 𝑥 ran 𝐹
109nfpw 4329 . . 3 𝑥𝒫 ran 𝐹
114, 7, 10nfif 4272 . 2 𝑥if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ran 𝐹)
121, 11nfcxfr 2905 1 𝑥(𝐹''''𝐴)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2894  ifcif 4243  𝒫 cpw 4315   cuni 4594   class class class wbr 4809  ran crn 5278  cio 6029   defAt wdfat 41796  ''''cafv2 41888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-iota 6031  df-fun 6070  df-dfat 41799  df-afv2 41889
This theorem is referenced by:  csbafv212g  41899
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