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| Mirrors > Home > MPE Home > Th. List > fmptdf | Structured version Visualization version GIF version | ||
| Description: A version of fmptd 7134 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 412 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶)) |
| 4 | 1, 3 | ralrimi 3257 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) |
| 5 | fmptdf.3 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 6 | 5 | fmpt 7130 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶) |
| 7 | 4, 6 | sylib 218 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 ↦ cmpt 5225 ⟶wf 6557 |
| 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-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 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-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: elrspunidl 33456 gsumesum 34060 voliune 34230 sdclem2 37749 fmptd2f 45240 limsupubuzmpt 45734 xlimmnfmpt 45858 xlimpnfmpt 45859 cncfiooicclem1 45908 stoweidlem35 46050 stoweidlem42 46057 stoweidlem48 46063 stirlinglem8 46096 sge0revalmpt 46393 sge0gerpmpt 46417 sge0ssrempt 46420 sge0ltfirpmpt 46423 sge0lempt 46425 sge0splitmpt 46426 sge0ss 46427 sge0rernmpt 46437 sge0lefimpt 46438 sge0clmpt 46440 sge0ltfirpmpt2 46441 sge0isummpt 46445 sge0xadd 46450 sge0fsummptf 46451 sge0snmptf 46452 sge0ge0mpt 46453 sge0repnfmpt 46454 sge0pnffigtmpt 46455 sge0gtfsumgt 46458 sge0pnfmpt 46460 meadjiun 46481 meaiunlelem 46483 omeiunle 46532 omeiunlempt 46535 opnvonmbllem1 46647 hoimbl2 46680 vonhoire 46687 vonn0ioo2 46705 vonn0icc2 46707 issmfdmpt 46763 smfconst 46764 smfadd 46780 smfpimcclem 46822 smflimmpt 46825 smflimsuplem2 46836 gsumsplit2f 48096 fsuppmptdmf 48294 |
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