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Mirrors > Home > MPE Home > Th. List > fmptdf | Structured version Visualization version GIF version |
Description: A version of fmptd 6548 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 449 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶)) |
4 | 1, 3 | ralrimi 3095 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) |
5 | fmptdf.3 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | fmpt 6544 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶) |
7 | 4, 6 | sylib 208 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 Ⅎwnf 1857 ∈ wcel 2139 ∀wral 3050 ↦ cmpt 4881 ⟶wf 6045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-fv 6057 |
This theorem is referenced by: gsumesum 30430 voliune 30601 sdclem2 33851 fmptd2f 39941 limsupubuzmpt 40454 xlimmnfmpt 40572 xlimpnfmpt 40573 cncfiooicclem1 40609 dvnprodlem1 40664 stoweidlem35 40755 stoweidlem42 40762 stoweidlem48 40768 stirlinglem8 40801 sge0z 41095 sge0revalmpt 41098 sge0f1o 41102 sge0gerpmpt 41122 sge0ssrempt 41125 sge0ltfirpmpt 41128 sge0lempt 41130 sge0splitmpt 41131 sge0ss 41132 sge0rernmpt 41142 sge0lefimpt 41143 sge0clmpt 41145 sge0ltfirpmpt2 41146 sge0isummpt 41150 sge0xadd 41155 sge0fsummptf 41156 sge0snmptf 41157 sge0ge0mpt 41158 sge0repnfmpt 41159 sge0pnffigtmpt 41160 sge0gtfsumgt 41163 sge0pnfmpt 41165 meadjiun 41186 meaiunlelem 41188 omeiunle 41237 omeiunlempt 41240 opnvonmbllem1 41352 hoimbl2 41385 vonhoire 41392 vonn0ioo2 41410 vonn0icc2 41412 pimgtmnf 41438 issmfdmpt 41463 smfconst 41464 smfadd 41479 smfpimcclem 41519 smflimmpt 41522 smfsupmpt 41527 smfinfmpt 41531 smflimsuplem2 41533 gsumsplit2f 42653 fsuppmptdmf 42672 |
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