Theorem nfafv2 47988

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6898. To prove a deduction version of this analogous to nffvd 6900 is not easily possible because a deduction version of nfdfat 47897 cannot be shown easily. (Contributed by AV, 4-Sep-2022.)

Hypotheses

Ref	Expression
nfafv2.1	⊢ Ⅎ𝑥𝐹
nfafv2.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfafv2	⊢ Ⅎ𝑥(𝐹''''𝐴)

Proof of Theorem nfafv2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-afv2 47979	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ∪ ran 𝐹)
2		nfafv2.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nfafv2.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfdfat 47897	. . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
5		nfcv 2928	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	3, 2, 5	nfbr 5163	. . . 4 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
7	6	nfiotaw 6503	. . 3 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
8	2	nfrn 5947	. . . . 5 ⊢ Ⅎ𝑥ran 𝐹
9	8	nfuni 4884	. . . 4 ⊢ Ⅎ𝑥∪ ran 𝐹
10	9	nfpw 4586	. . 3 ⊢ Ⅎ𝑥𝒫 ∪ ran 𝐹
11	4, 7, 10	nfif 4523	. 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ∪ ran 𝐹)
12	1, 11	nfcxfr 2926	1 ⊢ Ⅎ𝑥(𝐹''''𝐴)

This proof depends on syntax axioms:  Ⅎwnfc 2913  ifcif 4492  𝒫 cpw 4567  ∪ cuni 4877   class class class wbr 5114  ran crn 5667  ℩cio 6497   defAt wdfat 47886  ''''cafv2 47978

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-iota 6499  df-fun 6545  df-dfat 47889  df-afv2 47979

This theorem is used by:  csbafv212g  47989