Theorem nfafv2 47176

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6897. To prove a deduction version of this analogous to nffvd 6899 is not easily possible because a deduction version of nfdfat 47085 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 47167	. 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 47085	. . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
5		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	3, 2, 5	nfbr 5172	. . . 4 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
7	6	nfiotaw 6499	. . 3 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
8	2	nfrn 5945	. . . . 5 ⊢ Ⅎ𝑥ran 𝐹
9	8	nfuni 4896	. . . 4 ⊢ Ⅎ𝑥∪ ran 𝐹
10	9	nfpw 4601	. . 3 ⊢ Ⅎ𝑥𝒫 ∪ ran 𝐹
11	4, 7, 10	nfif 4538	. 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ∪ ran 𝐹)
12	1, 11	nfcxfr 2895	1 ⊢ Ⅎ𝑥(𝐹''''𝐴)

Syntax hints:  Ⅎwnfc 2882  ifcif 4507  𝒫 cpw 4582  ∪ cuni 4889   class class class wbr 5125  ran crn 5668  ℩cio 6493   defAt wdfat 47074  ''''cafv2 47166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-iota 6495  df-fun 6544  df-dfat 47077  df-afv2 47167

This theorem is referenced by:  csbafv212g  47177