Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 43765 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | iotaex 6304 | . . . 4 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V | |
3 | 2 | a1i 11 | . . 3 ⊢ (ran 𝐹 ∈ 𝑉 → (℩𝑥𝐴𝐹𝑥) ∈ V) |
4 | uniexg 7446 | . . . 4 ⊢ (ran 𝐹 ∈ 𝑉 → ∪ ran 𝐹 ∈ V) | |
5 | 4 | pwexd 5245 | . . 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 2894 | 1 ⊢ (ran 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 Vcvv 3441 ifcif 4425 𝒫 cpw 4497 ∪ cuni 4800 class class class wbr 5030 ran crn 5520 ℩cio 6281 defAt wdfat 43672 ''''cafv2 43764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-uni 4801 df-iota 6283 df-afv2 43765 |
This theorem is referenced by: fexafv2ex 43776 frnvafv2v 43792 |
Copyright terms: Public domain | W3C validator |