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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 45515 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | iotaex 6474 | . . . 4 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V | |
3 | 2 | a1i 11 | . . 3 ⊢ (ran 𝐹 ∈ 𝑉 → (℩𝑥𝐴𝐹𝑥) ∈ V) |
4 | uniexg 7682 | . . . 4 ⊢ (ran 𝐹 ∈ 𝑉 → ∪ ran 𝐹 ∈ V) | |
5 | 4 | pwexd 5339 | . . 3 ⊢ (ran 𝐹 ∈ 𝑉 → 𝒫 ∪ ran 𝐹 ∈ V) |
6 | 3, 5 | ifcld 4537 | . 2 ⊢ (ran 𝐹 ∈ 𝑉 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) ∈ V) |
7 | 1, 6 | eqeltrid 2842 | 1 ⊢ (ran 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3448 ifcif 4491 𝒫 cpw 4565 ∪ cuni 4870 class class class wbr 5110 ran crn 5639 ℩cio 6451 defAt wdfat 45422 ''''cafv2 45514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-uni 4871 df-iota 6453 df-afv2 45515 |
This theorem is referenced by: fexafv2ex 45526 fcdmvafv2v 45542 |
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