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Mirrors > Home > MPE Home > Th. List > fmptdf | Structured version Visualization version GIF version |
Description: A version of fmptd 7115 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 3253 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) |
5 | fmptdf.3 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | fmpt 7111 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶) |
7 | 4, 6 | sylib 217 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1784 ∈ wcel 2105 ∀wral 3060 ↦ cmpt 5231 ⟶wf 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-fun 6545 df-fn 6546 df-f 6547 |
This theorem is referenced by: elrspunidl 32986 gsumesum 33521 voliune 33691 sdclem2 37074 fmptd2f 44396 limsupubuzmpt 44894 xlimmnfmpt 45018 xlimpnfmpt 45019 cncfiooicclem1 45068 dvnprodlem1 45121 stoweidlem35 45210 stoweidlem42 45217 stoweidlem48 45223 stirlinglem8 45256 sge0revalmpt 45553 sge0f1o 45557 sge0gerpmpt 45577 sge0ssrempt 45580 sge0ltfirpmpt 45583 sge0lempt 45585 sge0splitmpt 45586 sge0ss 45587 sge0rernmpt 45597 sge0lefimpt 45598 sge0clmpt 45600 sge0ltfirpmpt2 45601 sge0isummpt 45605 sge0xadd 45610 sge0fsummptf 45611 sge0snmptf 45612 sge0ge0mpt 45613 sge0repnfmpt 45614 sge0pnffigtmpt 45615 sge0gtfsumgt 45618 sge0pnfmpt 45620 meadjiun 45641 meaiunlelem 45643 omeiunle 45692 omeiunlempt 45695 opnvonmbllem1 45807 hoimbl2 45840 vonhoire 45847 vonn0ioo2 45865 vonn0icc2 45867 issmfdmpt 45923 smfconst 45924 smfadd 45940 smfpimcclem 45982 smflimmpt 45985 smfinfmpt 45994 smflimsuplem2 45996 gsumsplit2f 47017 fsuppmptdmf 47220 |
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