mbfmf - Mathbox for Thierry Arnoux

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Theorem mbfmf 31513

Description: A measurable function as a function with domain and codomain. (Contributed by Thierry Arnoux, 25-Jan-2017.)

**Hypotheses**
Ref	Expression
mbfmf.1	⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
mbfmf.2	⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
mbfmf.3	⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))

**Assertion**
Ref	Expression
mbfmf	⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇)

Proof of Theorem mbfmf Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfmf.3	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))
2		mbfmf.1	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
3		mbfmf.2	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
4	2, 3	ismbfm 31510	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆)))
5	1, 4	mpbid 234	. . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆))
6	5	simpld 497	. 2 ⊢ (𝜑 → 𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆))
7		elmapi 8427	. 2 ⊢ (𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆) → 𝐹:∪ 𝑆⟶∪ 𝑇)
8	6, 7	syl 17	1 ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 ∪ cuni 4837 ^◡ccnv 5553 ran crn 5555 “ cima 5557 ⟶wf 6350 (class class class)co 7155 ↑_m cmap 8405 sigAlgebracsiga 31367 MblFnMcmbfm 31508

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 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460

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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-map 8407 df-mbfm 31509

This theorem is referenced by: imambfm 31520 mbfmco 31522 mbfmco2 31523 mbfmvolf 31524 sibff 31594 sitgclg 31600 orvcval4 31718

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