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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 24220 | . . . 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 3182 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) |
6 | dmmptg 6091 | . . . . 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 6495 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℂ ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)) |
9 | 3, 8 | mpbid 234 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
10 | 9 | fvmptelrn 6872 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ↦ cmpt 5139 dom cdm 5550 ⟶wf 6346 ℂcc 10529 MblFncmbf 24209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-pm 8403 df-mbf 24214 |
This theorem is referenced by: mbfss 24241 mbfneg 24245 mbfmulc2 24258 mbflim 24263 itgcnlem 24384 itgcnval 24394 itgre 24395 itgim 24396 iblneg 24397 itgneg 24398 iblss 24399 iblss2 24400 ibladd 24415 iblsub 24416 itgadd 24419 itgsub 24420 itgfsum 24421 iblabs 24423 iblabsr 24424 iblmulc2 24425 itgmulc2 24428 itgabs 24429 itgsplit 24430 bddmulibl 24433 itgcn 24437 ditgswap 24451 ditgsplitlem 24452 ftc1a 24628 ibladdnc 34943 itgaddnc 34946 iblsubnc 34947 itgsubnc 34948 iblabsnc 34950 iblmulc2nc 34951 itgmulc2nc 34954 itgabsnc 34955 |
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