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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 43483 | . 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) | |
2 | iotaex 6328 | . 2 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V | |
3 | 1, 2 | eqeltrdi 2920 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 Vcvv 3491 class class class wbr 5059 ℩cio 6305 defAt wdfat 43389 ''''cafv2 43481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-nul 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-uni 4832 df-iota 6307 df-afv2 43482 |
This theorem is referenced by: dfatbrafv2b 43518 fnbrafv2b 43521 dfatdmfcoafv2 43527 dfatcolem 43528 |
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