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Mirrors > Home > MPE Home > Th. List > fmptdf | Structured version Visualization version GIF version |
Description: A version of fmptd 6931 using bound-variable hypothesis instead of a distinct variable condition for 𝜑. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
fmptdf.1 | ⊢ Ⅎ𝑥𝜑 |
fmptdf.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
fmptdf.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
fmptdf | ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptdf.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | fmptdf.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
3 | 2 | ex 416 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶)) |
4 | 1, 3 | ralrimi 3137 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) |
5 | fmptdf.3 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | fmpt 6927 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶) |
7 | 4, 6 | sylib 221 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 ∀wral 3061 ↦ cmpt 5135 ⟶wf 6376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-fun 6382 df-fn 6383 df-f 6384 |
This theorem is referenced by: elrspunidl 31320 gsumesum 31739 voliune 31909 sdclem2 35637 fmptd2f 42451 limsupubuzmpt 42935 xlimmnfmpt 43059 xlimpnfmpt 43060 cncfiooicclem1 43109 dvnprodlem1 43162 stoweidlem35 43251 stoweidlem42 43258 stoweidlem48 43264 stirlinglem8 43297 sge0revalmpt 43591 sge0f1o 43595 sge0gerpmpt 43615 sge0ssrempt 43618 sge0ltfirpmpt 43621 sge0lempt 43623 sge0splitmpt 43624 sge0ss 43625 sge0rernmpt 43635 sge0lefimpt 43636 sge0clmpt 43638 sge0ltfirpmpt2 43639 sge0isummpt 43643 sge0xadd 43648 sge0fsummptf 43649 sge0snmptf 43650 sge0ge0mpt 43651 sge0repnfmpt 43652 sge0pnffigtmpt 43653 sge0gtfsumgt 43656 sge0pnfmpt 43658 meadjiun 43679 meaiunlelem 43681 omeiunle 43730 omeiunlempt 43733 opnvonmbllem1 43845 hoimbl2 43878 vonhoire 43885 vonn0ioo2 43903 vonn0icc2 43905 pimgtmnf 43931 issmfdmpt 43956 smfconst 43957 smfadd 43972 smfpimcclem 44012 smflimmpt 44015 smfsupmpt 44020 smfinfmpt 44024 smflimsuplem2 44026 gsumsplit2f 45047 fsuppmptdmf 45390 |
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