Theorem nfafv2 47168

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6917. To prove a deduction version of this analogous to nffvd 6919 is not easily possible because a deduction version of nfdfat 47077 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 47159	. 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 47077	. . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
5		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	3, 2, 5	nfbr 5195	. . . 4 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
7	6	nfiotaw 6520	. . 3 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
8	2	nfrn 5966	. . . . 5 ⊢ Ⅎ𝑥ran 𝐹
9	8	nfuni 4919	. . . 4 ⊢ Ⅎ𝑥∪ ran 𝐹
10	9	nfpw 4624	. . 3 ⊢ Ⅎ𝑥𝒫 ∪ ran 𝐹
11	4, 7, 10	nfif 4561	. 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ∪ ran 𝐹)
12	1, 11	nfcxfr 2901	1 ⊢ Ⅎ𝑥(𝐹''''𝐴)

Syntax hints:  Ⅎwnfc 2888  ifcif 4531  𝒫 cpw 4605  ∪ cuni 4912   class class class wbr 5148  ran crn 5690  ℩cio 6514   defAt wdfat 47066  ''''cafv2 47158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-dfat 47069  df-afv2 47159

This theorem is referenced by:  csbafv212g  47169