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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfatafv2ex | Structured version Visualization version GIF version |
Description: The alternate function value at a class 𝐴 is always a set if the function/class 𝐹 is defined at 𝐴. (Contributed by AV, 6-Sep-2022.) |
Ref | Expression |
---|---|
dfatafv2ex | ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfatafv2iota 43429 | . 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) | |
2 | iotaex 6335 | . 2 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V | |
3 | 1, 2 | eqeltrdi 2921 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 ℩cio 6312 defAt wdfat 43335 ''''cafv2 43427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-uni 4839 df-iota 6314 df-afv2 43428 |
This theorem is referenced by: dfatbrafv2b 43464 fnbrafv2b 43467 dfatdmfcoafv2 43473 dfatcolem 43474 |
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