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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afv2ex | Structured version Visualization version GIF version |
Description: The alternate function value is always a set if the range of the function is a set. (Contributed by AV, 2-Sep-2022.) |
Ref | Expression |
---|---|
afv2ex | ⊢ (ran 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-afv2 46463 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | iotaex 6507 | . . . 4 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V | |
3 | 2 | a1i 11 | . . 3 ⊢ (ran 𝐹 ∈ 𝑉 → (℩𝑥𝐴𝐹𝑥) ∈ V) |
4 | uniexg 7724 | . . . 4 ⊢ (ran 𝐹 ∈ 𝑉 → ∪ ran 𝐹 ∈ V) | |
5 | 4 | pwexd 5368 | . . 3 ⊢ (ran 𝐹 ∈ 𝑉 → 𝒫 ∪ ran 𝐹 ∈ V) |
6 | 3, 5 | ifcld 4567 | . 2 ⊢ (ran 𝐹 ∈ 𝑉 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) ∈ V) |
7 | 1, 6 | eqeltrid 2829 | 1 ⊢ (ran 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3466 ifcif 4521 𝒫 cpw 4595 ∪ cuni 4900 class class class wbr 5139 ran crn 5668 ℩cio 6484 defAt wdfat 46370 ''''cafv2 46462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-uni 4901 df-iota 6486 df-afv2 46463 |
This theorem is referenced by: fexafv2ex 46474 fcdmvafv2v 46490 |
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