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Theorem nfafv2 43283
Description: Bound-variable hypothesis builder for function value, analogous to nffv 6677. To prove a deduction version of this analogous to nffvd 6679 is not easily possible because a deduction version of nfdfat 43192 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 43274 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ran 𝐹)
2 nfafv2.1 . . . 4 𝑥𝐹
3 nfafv2.2 . . . 4 𝑥𝐴
42, 3nfdfat 43192 . . 3 𝑥 𝐹 defAt 𝐴
5 nfcv 2982 . . . . 5 𝑥𝑦
63, 2, 5nfbr 5110 . . . 4 𝑥 𝐴𝐹𝑦
76nfiota 6318 . . 3 𝑥(℩𝑦𝐴𝐹𝑦)
82nfrn 5823 . . . . 5 𝑥ran 𝐹
98nfuni 4844 . . . 4 𝑥 ran 𝐹
109nfpw 4558 . . 3 𝑥𝒫 ran 𝐹
114, 7, 10nfif 4499 . 2 𝑥if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ran 𝐹)
121, 11nfcxfr 2980 1 𝑥(𝐹''''𝐴)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2966  ifcif 4470  𝒫 cpw 4542   cuni 4837   class class class wbr 5063  ran crn 5555  cio 6310   defAt wdfat 43181  ''''cafv2 43273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-iota 6312  df-fun 6354  df-dfat 43184  df-afv2 43274
This theorem is referenced by:  csbafv212g  43284
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