Theorem nfafv2 47406

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6842. To prove a deduction version of this analogous to nffvd 6844 is not easily possible because a deduction version of nfdfat 47315 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 47397	. 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 47315	. . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
5		nfcv 2896	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	3, 2, 5	nfbr 5143	. . . 4 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
7	6	nfiotaw 6450	. . 3 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
8	2	nfrn 5899	. . . . 5 ⊢ Ⅎ𝑥ran 𝐹
9	8	nfuni 4868	. . . 4 ⊢ Ⅎ𝑥∪ ran 𝐹
10	9	nfpw 4571	. . 3 ⊢ Ⅎ𝑥𝒫 ∪ ran 𝐹
11	4, 7, 10	nfif 4508	. 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ∪ ran 𝐹)
12	1, 11	nfcxfr 2894	1 ⊢ Ⅎ𝑥(𝐹''''𝐴)

Syntax hints:  Ⅎwnfc 2881  ifcif 4477  𝒫 cpw 4552  ∪ cuni 4861   class class class wbr 5096  ran crn 5623  ℩cio 6444   defAt wdfat 47304  ''''cafv2 47396

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-iota 6446  df-fun 6492  df-dfat 47307  df-afv2 47397

This theorem is referenced by:  csbafv212g  47407