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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 7003 | 1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ↦ cmpt 5232 ‘cfv 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 |
This theorem is referenced by: isf32lem9 10356 axcc2lem 10431 caucvg 15625 ismre 17534 mrisval 17574 frmdup1 18745 frmdup2 18746 qusghm 19129 pmtrfval 19318 odf1 19430 vrgpfval 19634 dprdz 19900 dmdprdsplitlem 19907 dprd2dlem2 19910 dprd2dlem1 19911 dprd2da 19912 ablfac1a 19939 ablfac1b 19940 ablfac1eu 19943 ipdir 21192 ipass 21198 isphld 21207 istopon 22414 qustgpopn 23624 qustgplem 23625 tcphcph 24754 cmvth 25508 mvth 25509 dvle 25524 lhop1 25531 dvfsumlem3 25545 pige3ALT 26029 fsumdvdscom 26689 logfacbnd3 26726 dchrptlem1 26767 dchrptlem2 26768 lgsdchrval 26857 dchrisumlem3 26994 dchrisum0flblem1 27011 dchrisum0fno1 27014 dchrisum0lem1b 27018 dchrisum0lem2a 27020 dchrisum0lem2 27021 logsqvma2 27046 log2sumbnd 27047 measdivcst 33253 measdivcstALTV 33254 mrexval 34523 mexval 34524 mdvval 34526 msubvrs 34582 mthmval 34597 gg-cmvth 35212 f1omptsnlem 36265 upixp 36645 ismrer1 36754 frlmsnic 41158 fsuppind 41210 uzmptshftfval 43153 amgmwlem 47897 amgmlemALT 47898 |
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