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| Mirrors > Home > MPE Home > Th. List > fvmpt3i | Structured version Visualization version GIF version | ||
| Description: Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.) |
| Ref | Expression |
|---|---|
| fvmpt3.a | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
| fvmpt3.b | ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
| fvmpt3i.c | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| fvmpt3i | ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvmpt3.a | . 2 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
| 2 | fvmpt3.b | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
| 3 | fvmpt3i.c | . . 3 ⊢ 𝐵 ∈ V | |
| 4 | 3 | a1i 11 | . 2 ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ V) |
| 5 | 1, 2, 4 | fvmpt3 7020 | 1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ↦ cmpt 5225 ‘cfv 6561 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: isf32lem9 10401 axcc2lem 10476 caucvg 15715 ismre 17633 mrisval 17673 frmdup1 18877 frmdup2 18878 qusghm 19273 pmtrfval 19468 odf1 19580 vrgpfval 19784 dprdz 20050 dmdprdsplitlem 20057 dprd2dlem2 20060 dprd2dlem1 20061 dprd2da 20062 ablfac1a 20089 ablfac1b 20090 ablfac1eu 20093 ipdir 21657 ipass 21663 isphld 21672 istopon 22918 qustgpopn 24128 qustgplem 24129 tcphcph 25271 cmvth 26029 cmvthOLD 26030 mvth 26031 dvle 26046 lhop1 26053 dvfsumlem3 26069 pige3ALT 26562 fsumdvdscom 27228 logfacbnd3 27267 dchrptlem1 27308 dchrptlem2 27309 lgsdchrval 27398 dchrisumlem3 27535 dchrisum0flblem1 27552 dchrisum0fno1 27555 dchrisum0lem1b 27559 dchrisum0lem2a 27561 dchrisum0lem2 27562 logsqvma2 27587 log2sumbnd 27588 zringfrac 33582 measdivcst 34225 measdivcstALTV 34226 mrexval 35506 mexval 35507 mdvval 35509 msubvrs 35565 mthmval 35580 weiunlem2 36464 f1omptsnlem 37337 upixp 37736 ismrer1 37845 frlmsnic 42550 fsuppind 42600 uzmptshftfval 44365 tposideq 48788 fucocolem2 49049 amgmwlem 49321 amgmlemALT 49322 |
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