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Mirrors > Home > MPE Home > Th. List > mbfmptcl | Structured version Visualization version GIF version |
Description: Lemma for the MblFn predicate applied to a mapping operation. (Contributed by Mario Carneiro, 11-Aug-2014.) |
Ref | Expression |
---|---|
mbfmptcl.1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
mbfmptcl.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
Ref | Expression |
---|---|
mbfmptcl | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbfmptcl.1 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) | |
2 | mbff 25642 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn → (𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℂ) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℂ) |
4 | mbfmptcl.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
5 | 4 | ralrimiva 3136 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) |
6 | dmmptg 6245 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
8 | 7 | feq2d 6706 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℂ ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)) |
9 | 3, 8 | mpbid 231 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
10 | 9 | fvmptelcdm 7119 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ↦ cmpt 5228 dom cdm 5674 ⟶wf 6542 ℂcc 11147 MblFncmbf 25631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-pm 8850 df-mbf 25636 |
This theorem is referenced by: mbfss 25663 mbfneg 25667 mbfmulc2 25680 mbflim 25685 itgcnlem 25807 itgcnval 25817 itgre 25818 itgim 25819 iblneg 25820 itgneg 25821 iblss 25822 iblss2 25823 ibladd 25838 iblsub 25839 itgadd 25842 itgsub 25843 itgfsum 25844 iblabs 25846 iblabsr 25847 iblmulc2 25848 itgmulc2 25851 itgabs 25852 itgsplit 25853 bddmulibl 25856 itgcn 25862 ditgswap 25876 ditgsplitlem 25877 ftc1a 26060 ibladdnc 37391 itgaddnc 37394 iblsubnc 37395 itgsubnc 37396 iblabsnc 37398 iblmulc2nc 37399 itgmulc2nc 37402 itgabsnc 37403 |
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