Theorem nfafv2 46385

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6901. To prove a deduction version of this analogous to nffvd 6903 is not easily possible because a deduction version of nfdfat 46294 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 46376	. 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 46294	. . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
5		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	3, 2, 5	nfbr 5195	. . . 4 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
7	6	nfiotaw 6499	. . 3 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
8	2	nfrn 5951	. . . . 5 ⊢ Ⅎ𝑥ran 𝐹
9	8	nfuni 4915	. . . 4 ⊢ Ⅎ𝑥∪ ran 𝐹
10	9	nfpw 4621	. . 3 ⊢ Ⅎ𝑥𝒫 ∪ ran 𝐹
11	4, 7, 10	nfif 4558	. 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ∪ ran 𝐹)
12	1, 11	nfcxfr 2900	1 ⊢ Ⅎ𝑥(𝐹''''𝐴)

Syntax hints:  Ⅎwnfc 2882  ifcif 4528  𝒫 cpw 4602  ∪ cuni 4908   class class class wbr 5148  ran crn 5677  ℩cio 6493   defAt wdfat 46283  ''''cafv2 46375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-dfat 46286  df-afv2 46376

This theorem is referenced by:  csbafv212g  46386