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Mirrors > Home > ILE Home > Th. List > fvmptg | GIF version |
Description: Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fvmptg.1 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
fvmptg.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
Ref | Expression |
---|---|
fvmptg | ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐹‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2089 | . 2 ⊢ 𝐶 = 𝐶 | |
2 | fvmptg.1 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
3 | 2 | eqeq2d 2100 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶)) |
4 | eqeq1 2095 | . . 3 ⊢ (𝑦 = 𝐶 → (𝑦 = 𝐶 ↔ 𝐶 = 𝐶)) | |
5 | moeq 2791 | . . . 4 ⊢ ∃*𝑦 𝑦 = 𝐵 | |
6 | 5 | a1i 9 | . . 3 ⊢ (𝑥 ∈ 𝐷 → ∃*𝑦 𝑦 = 𝐵) |
7 | fvmptg.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
8 | df-mpt 3907 | . . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐷 ∧ 𝑦 = 𝐵)} | |
9 | 7, 8 | eqtri 2109 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐷 ∧ 𝑦 = 𝐵)} |
10 | 3, 4, 6, 9 | fvopab3ig 5391 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐶 = 𝐶 → (𝐹‘𝐴) = 𝐶)) |
11 | 1, 10 | mpi 15 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1290 ∈ wcel 1439 ∃*wmo 1950 {copab 3904 ↦ cmpt 3905 ‘cfv 5028 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-sbc 2842 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-iota 4993 df-fun 5030 df-fv 5036 |
This theorem is referenced by: fvmpt 5394 fvmpts 5395 fvmpt3 5396 fvmpt2 5399 f1mpt 5564 fnofval 5879 caofinvl 5891 1stvalg 5927 2ndvalg 5928 brtpos2 6030 rdgon 6165 frec0g 6176 freccllem 6181 frecfcllem 6183 frecsuclem 6185 sucinc 6220 sucinc2 6221 omcl 6236 oeicl 6237 oav2 6238 omv2 6240 fvdiagfn 6464 djulclr 6795 djurclr 6796 djulcl 6797 djurcl 6798 djulclb 6801 djur 6811 fodjuomnilemm 6855 fodjuomnilem0 6856 nnnninf 6860 cardval3ex 6867 ceilqval 9767 frec2uzzd 9861 frec2uzsucd 9862 monoord2 9959 iseqf1olemqval 9970 iseqf1olemqk 9977 seq3f1olemqsum 9983 seq3f1oleml 9986 seq3f1o 9987 iseqdistr 9999 seq3distr 10000 ser3le 10007 hashinfom 10240 hashennn 10242 cjval 10333 reval 10337 imval 10338 cvg1nlemcau 10471 cvg1nlemres 10472 absval 10488 resqrexlemglsq 10509 resqrexlemga 10510 climmpt 10742 climle 10776 climcvg1nlem 10792 isummolem3 10824 isummolem2a 10825 zisum 10828 fisum 10832 fsum3 10833 fsumf1o 10836 fisumser 10844 fsumcl2lem 10846 fsumadd 10854 sumsnf 10857 isumadd 10879 fsumrev 10891 fsumshft 10892 fsummulc2 10896 iserabs 10923 isumlessdc 10944 divcnv 10945 trireciplem 10948 trirecip 10949 expcnvap0 10950 expcnvre 10951 expcnv 10952 explecnv 10953 geolim 10959 geolim2 10960 geo2lim 10964 geoisum 10965 geoisumr 10966 geoisum1 10967 geoisum1c 10968 cvgratz 10980 mertenslem2 10984 mertensabs 10985 eftvalcn 11001 efval 11005 efcvgfsum 11011 ege2le3 11015 efcj 11017 eftlub 11034 efgt1p2 11039 eflegeo 11046 sinval 11047 cosval 11048 tanvalap 11053 eirraplem 11118 phival 11521 crth 11532 phimullem 11533 strnfvnd 11568 topnvalg 11718 toponsspwpwg 11774 tgval 11803 cldval 11853 ntrfval 11854 clsfval 11855 mulc1cncf 11911 djucllem 11966 nnsf 12161 peano3nninf 12163 nninfalllemn 12164 nninfself 12171 nninfsellemeqinf 12174 |
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