Theorem nfafv2 48109

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6889. To prove a deduction version of this analogous to nffvd 6891 is not easily possible because a deduction version of nfdfat 48018 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 48100	. 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 48018	. . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
5		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	3, 2, 5	nfbr 5152	. . . 4 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
7	6	nfiotaw 6493	. . 3 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
8	2	nfrn 5936	. . . . 5 ⊢ Ⅎ𝑥ran 𝐹
9	8	nfuni 4874	. . . 4 ⊢ Ⅎ𝑥∪ ran 𝐹
10	9	nfpw 4576	. . 3 ⊢ Ⅎ𝑥𝒫 ∪ ran 𝐹
11	4, 7, 10	nfif 4513	. 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ∪ ran 𝐹)
12	1, 11	nfcxfr 2920	1 ⊢ Ⅎ𝑥(𝐹''''𝐴)

This proof depends on syntax axioms:  Ⅎwnfc 2907  ifcif 4482  𝒫 cpw 4557  ∪ cuni 4867   class class class wbr 5103  ran crn 5656  ℩cio 6487   defAt wdfat 48007  ''''cafv2 48099

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

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 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-iota 6489  df-fun 6535  df-dfat 48010  df-afv2 48100

This theorem is used by:  csbafv212g  48110