mbfmfun - Mathbox for Thierry Arnoux

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Theorem mbfmfun 33715

Description: A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)

**Hypothesis**
Ref	Expression
mbfmfun.1	⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)

**Assertion**
Ref	Expression
mbfmfun	⊢ (𝜑 → Fun 𝐹)

Proof of Theorem mbfmfun Dummy variables 𝑡 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfmfun.1	. 2 ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)
2		elunirnmbfm 33714	. . 3 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑_m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (^◡𝐹 “ 𝑥) ∈ 𝑠))
3	2	biimpi 215	. 2 ⊢ (𝐹 ∈ ∪ ran MblFnM → ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑_m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (^◡𝐹 “ 𝑥) ∈ 𝑠))
4		elmapfun 8866	. . . . 5 ⊢ (𝐹 ∈ (∪ 𝑡 ↑_m ∪ 𝑠) → Fun 𝐹)
5	4	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (∪ 𝑡 ↑_m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (^◡𝐹 “ 𝑥) ∈ 𝑠) → Fun 𝐹)
6	5	rexlimivw 3150	. . 3 ⊢ (∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑_m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (^◡𝐹 “ 𝑥) ∈ 𝑠) → Fun 𝐹)
7	6	rexlimivw 3150	. 2 ⊢ (∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑_m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (^◡𝐹 “ 𝑥) ∈ 𝑠) → Fun 𝐹)
8	1, 3, 7	3syl 18	1 ⊢ (𝜑 → Fun 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 ∪ cuni 4908 ^◡ccnv 5675 ran crn 5677 “ cima 5679 Fun wfun 6537 (class class class)co 7412 ↑_m cmap 8826 sigAlgebracsiga 33570 MblFnMcmbfm 33711

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-map 8828 df-mbfm 33712

This theorem is referenced by: orvcval4 33923

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