Theorem nfafv2 48433

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6906. To prove a deduction version of this analogous to nffvd 6908 is not easily possible because a deduction version of nfdfat 48342 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 48424	. 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 48342	. . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
5		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	3, 2, 5	nfbr 5155	. . . 4 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
7	6	nfiotaw 6505	. . 3 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
8	2	nfrn 5939	. . . . 5 ⊢ Ⅎ𝑥ran 𝐹
9	8	nfuni 4877	. . . 4 ⊢ Ⅎ𝑥∪ ran 𝐹
10	9	nfpw 4580	. . 3 ⊢ Ⅎ𝑥𝒫 ∪ ran 𝐹
11	4, 7, 10	nfif 4516	. 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ∪ ran 𝐹)
12	1, 11	nfcxfr 2924	1 ⊢ Ⅎ𝑥(𝐹''''𝐴)

This proof depends on syntax axioms:  Ⅎwnfc 2911  ifcif 4485  𝒫 cpw 4560  ∪ cuni 4870   class class class wbr 5106  ran crn 5656  ℩cio 6499   defAt wdfat 48332  ''''cafv2 48423

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-iota 6501  df-fun 6549  df-dfat 48335  df-afv2 48424

This theorem is used by:  csbafv212g  48434