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| Mirrors > Home > MPE Home > Th. List > fvmptg | Structured version Visualization version 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 2737 | . 2 ⊢ 𝐶 = 𝐶 | |
| 2 | fvmptg.1 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
| 3 | 2 | eqeq2d 2748 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶)) |
| 4 | eqeq1 2741 | . . 3 ⊢ (𝑦 = 𝐶 → (𝑦 = 𝐶 ↔ 𝐶 = 𝐶)) | |
| 5 | moeq 3713 | . . . 4 ⊢ ∃*𝑦 𝑦 = 𝐵 | |
| 6 | 5 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐷 → ∃*𝑦 𝑦 = 𝐵) |
| 7 | fvmptg.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
| 8 | df-mpt 5226 | . . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐷 ∧ 𝑦 = 𝐵)} | |
| 9 | 7, 8 | eqtri 2765 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐷 ∧ 𝑦 = 𝐵)} |
| 10 | 3, 4, 6, 9 | fvopab3ig 7012 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐶 = 𝐶 → (𝐹‘𝐴) = 𝐶)) |
| 11 | 1, 10 | mpi 20 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐹‘𝐴) = 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃*wmo 2538 {copab 5205 ↦ 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: fvmpti 7015 fvmpt 7016 fvmpt2f 7017 fvtresfn 7018 fvmpts 7019 fvmpt3 7020 fvmptd3 7039 fvmptss2 7042 f1mpt 7281 bropfvvvv 8117 tz7.44-3 8448 pw2f1olem 9116 wdom2d 9620 tz9.12lem3 9829 djurcl 9951 djur 9959 djuun 9966 cardval3 9992 cfval 10287 coftr 10313 fin1a2lem1 10440 fin1a2lem12 10451 axdc2lem 10488 pwcfsdom 10623 tskmval 10879 lsw 14602 swrdswrd 14743 trclfv 15039 relexpsucnnr 15064 dfrtrclrec2 15097 rtrclreclem2 15098 summolem2a 15751 prodmolem2a 15970 divsfval 17592 joinfval 18418 meetfval 18432 symgextfv 19436 symgextfve 19437 pmtrdifwrdel2lem1 19502 efgtf 19740 rrgsupp 20701 uvcvval 21806 ply1sclid 22291 submaval0 22586 m2detleiblem3 22635 m2detleiblem4 22636 maduval 22644 minmar1val0 22653 toponsspwpw 22928 cldval 23031 ntrfval 23032 clsfval 23033 opncldf3 23094 neifval 23107 lpfval 23146 islocfin 23525 kqfval 23731 stdbdxmet 24528 cmetcaulem 25322 bcth3 25365 itg2gt0 25795 ellimc2 25912 coe1termlem 26297 bdayval 27693 oldval 27893 clwlkclwwlkfo 30028 grpoinvfval 30541 grpodivfval 30553 nlfnval 31900 sigaval 34112 measval 34199 measdivcst 34225 measdivcstALTV 34226 probfinmeasbALTV 34431 ptpconn 35238 cvmsval 35271 ex-sategoelel12 35432 imageval 35931 fvimage 35932 tailfval 36373 tailval 36374 curfv 37607 heiborlem4 37821 lkrval 39089 cdleme31fv 40392 docavalN 41125 dochval 41353 mapdval 41630 hvmapval 41762 hvmapvalvalN 41763 hdmap1vallem 41799 hdmapval 41830 hgmapval 41889 mzpval 42743 mzpsubst 42759 pw2f1o2val 43051 refsum2cnlem1 45042 stoweidlem26 46041 stirlinglem8 46096 fourierdlem50 46171 caragenval 46508 fargshiftfv 47426 lincvalsc0 48338 linc0scn0 48340 linc1 48342 lincscm 48347 |
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