mbfmfun - Mathbox for Thierry Arnoux

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Theorem mbfmfun 33068

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 33067	. . 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 8842	. . . . 5 ⊢ (𝐹 ∈ (∪ 𝑡 ↑_m ∪ 𝑠) → Fun 𝐹)
5	4	adantr 481	. . . 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 396 ∈ wcel 2106 ∀wral 3060 ∃wrex 3069 ∪ cuni 4900 ^◡ccnv 5667 ran crn 5669 “ cima 5671 Fun wfun 6525 (class class class)co 7392 ↑_m cmap 8802 sigAlgebracsiga 32923 MblFnMcmbfm 33064

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5291 ax-nul 5298 ax-pow 5355 ax-pr 5419 ax-un 7707

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3474 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4991 df-br 5141 df-opab 5203 df-mpt 5224 df-id 5566 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-rn 5679 df-res 5680 df-ima 5681 df-iota 6483 df-fun 6533 df-fn 6534 df-f 6535 df-fv 6539 df-ov 7395 df-oprab 7396 df-mpo 7397 df-1st 7956 df-2nd 7957 df-map 8804 df-mbfm 33065

This theorem is referenced by: orvcval4 33276

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