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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfafv2 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for function value, analogous to nffv 6814. To prove a deduction version of this analogous to nffvd 6816 is not easily possible because a deduction version of nfdfat 44863 cannot be shown easily. (Contributed by AV, 4-Sep-2022.) |
Ref | Expression |
---|---|
nfafv2.1 | ⊢ Ⅎ𝑥𝐹 |
nfafv2.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfafv2 | ⊢ Ⅎ𝑥(𝐹''''𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-afv2 44945 | . 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 44863 | . . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴 |
5 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
6 | 3, 2, 5 | nfbr 5128 | . . . 4 ⊢ Ⅎ𝑥 𝐴𝐹𝑦 |
7 | 6 | nfiotaw 6414 | . . 3 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦) |
8 | 2 | nfrn 5873 | . . . . 5 ⊢ Ⅎ𝑥ran 𝐹 |
9 | 8 | nfuni 4851 | . . . 4 ⊢ Ⅎ𝑥∪ ran 𝐹 |
10 | 9 | nfpw 4558 | . . 3 ⊢ Ⅎ𝑥𝒫 ∪ ran 𝐹 |
11 | 4, 7, 10 | nfif 4495 | . 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (℩𝑦𝐴𝐹𝑦), 𝒫 ∪ ran 𝐹) |
12 | 1, 11 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥(𝐹''''𝐴) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2885 ifcif 4465 𝒫 cpw 4539 ∪ cuni 4844 class class class wbr 5081 ran crn 5601 ℩cio 6408 defAt wdfat 44852 ''''cafv2 44944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3306 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-iota 6410 df-fun 6460 df-dfat 44855 df-afv2 44945 |
This theorem is referenced by: csbafv212g 44955 |
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