Theorem mbfmfun 34254

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 34253	. . 3 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠))
3	2	biimpi 216	. 2 ⊢ (𝐹 ∈ ∪ ran MblFnM → ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠))
4		elmapfun 8906	. . . . 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 3151	. . 3 ⊢ (∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠) → Fun 𝐹)
7	6	rexlimivw 3151	. 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)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  ∪ cuni 4907  ^◡ccnv 5684  ran crn 5686   “ cima 5688  Fun wfun 6555  (class class class)co 7431   ↑_m cmap 8866  sigAlgebracsiga 34109  MblFnMcmbfm 34250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-mbfm 34251

This theorem is referenced by:  orvcval4  34463