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 44281 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | iotaex 6329 | . . . 4 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V | |
3 | 2 | a1i 11 | . . 3 ⊢ (ran 𝐹 ∈ 𝑉 → (℩𝑥𝐴𝐹𝑥) ∈ V) |
4 | uniexg 7496 | . . . 4 ⊢ (ran 𝐹 ∈ 𝑉 → ∪ ran 𝐹 ∈ V) | |
5 | 4 | pwexd 5256 | . . 3 ⊢ (ran 𝐹 ∈ 𝑉 → 𝒫 ∪ ran 𝐹 ∈ V) |
6 | 3, 5 | ifcld 4470 | . 2 ⊢ (ran 𝐹 ∈ 𝑉 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) ∈ V) |
7 | 1, 6 | eqeltrid 2838 | 1 ⊢ (ran 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3400 ifcif 4424 𝒫 cpw 4498 ∪ cuni 4806 class class class wbr 5040 ran crn 5536 ℩cio 6305 defAt wdfat 44188 ''''cafv2 44280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-un 7491 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-uni 4807 df-iota 6307 df-afv2 44281 |
This theorem is referenced by: fexafv2ex 44292 frnvafv2v 44308 |
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