mbfmf - Mathbox for Thierry Arnoux

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Theorem mbfmf 33867

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 33864	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆)))
5	1, 4	mpbid 231	. . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆))
6	5	simpld 494	. 2 ⊢ (𝜑 → 𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆))
7		elmapi 8861	. 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 395 ∈ wcel 2099 ∀wral 3057 ∪ cuni 4903 ^◡ccnv 5671 ran crn 5673 “ cima 5675 ⟶wf 6538 (class class class)co 7414 ↑_m cmap 8838 sigAlgebracsiga 33721 MblFnMcmbfm 33862

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 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-map 8840 df-mbfm 33863

This theorem is referenced by: imambfm 33876 mbfmco 33878 mbfmco2 33879 mbfmvolf 33880 sibff 33950 sitgclg 33956 orvcval4 34074

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