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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 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) | |
2 | mbff 23200 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn → (𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℂ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℂ) |
4 | mbfmptcl.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
5 | 4 | ralrimiva 2949 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) |
6 | dmmptg 5549 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
8 | 7 | feq2d 5944 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℂ ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)) |
9 | 3, 8 | mpbid 221 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
10 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
11 | 10 | fmpt 6289 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
12 | 9, 11 | sylibr 223 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ) |
13 | 12 | r19.21bi 2916 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ↦ cmpt 4643 dom cdm 5038 ⟶wf 5800 ℂcc 9813 MblFncmbf 23189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-pm 7747 df-mbf 23194 |
This theorem is referenced by: mbfss 23219 mbfneg 23223 mbfmulc2 23236 mbflim 23241 itgcnlem 23362 itgcnval 23372 itgre 23373 itgim 23374 iblneg 23375 itgneg 23376 iblss 23377 iblss2 23378 ibladd 23393 iblsub 23394 itgadd 23397 itgsub 23398 itgfsum 23399 iblabs 23401 iblabsr 23402 iblmulc2 23403 itgmulc2 23406 itgabs 23407 itgsplit 23408 bddmulibl 23411 itgcn 23415 ditgswap 23429 ditgsplitlem 23430 ftc1a 23604 ibladdnc 32637 itgaddnc 32640 iblsubnc 32641 itgsubnc 32642 iblabsnc 32644 iblmulc2nc 32645 itgmulc2nc 32648 itgabsnc 32649 |
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