Theorem nfafv2 47666

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6850. To prove a deduction version of this analogous to nffvd 6852 is not easily possible because a deduction version of nfdfat 47575 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 47657	. 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 47575	. . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
5		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	3, 2, 5	nfbr 5132	. . . 4 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
7	6	nfiotaw 6458	. . 3 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
8	2	nfrn 5907	. . . . 5 ⊢ Ⅎ𝑥ran 𝐹
9	8	nfuni 4857	. . . 4 ⊢ Ⅎ𝑥∪ ran 𝐹
10	9	nfpw 4560	. . 3 ⊢ Ⅎ𝑥𝒫 ∪ ran 𝐹
11	4, 7, 10	nfif 4497	. 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ∪ ran 𝐹)
12	1, 11	nfcxfr 2896	1 ⊢ Ⅎ𝑥(𝐹''''𝐴)

Syntax hints:  Ⅎwnfc 2883  ifcif 4466  𝒫 cpw 4541  ∪ cuni 4850   class class class wbr 5085  ran crn 5632  ℩cio 6452   defAt wdfat 47564  ''''cafv2 47656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6454  df-fun 6500  df-dfat 47567  df-afv2 47657

This theorem is referenced by:  csbafv212g  47667