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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afv2rnfveq | Structured version Visualization version GIF version |
Description: If the alternate function value is defined, i.e., in the range of the function, the alternate function value equals the function's value. (Contributed by AV, 3-Sep-2022.) |
Ref | Expression |
---|---|
afv2rnfveq | ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → (𝐹''''𝐴) = (𝐹‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfatafv2rnb 45771 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹) | |
2 | dfatafv2eqfv 45805 | . 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴)) | |
3 | 1, 2 | sylbir 234 | 1 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → (𝐹''''𝐴) = (𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ran crn 5671 ‘cfv 6533 defAt wdfat 45660 ''''cafv2 45752 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-iota 6485 df-fun 6535 df-fv 6541 df-dfat 45663 df-afv2 45753 |
This theorem is referenced by: afv2fv0 45809 |
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