Theorem mbfmfun 34266

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 34265	. . 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 8790	. . . . 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 3129	. . 3 ⊢ (∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠) → Fun 𝐹)
7	6	rexlimivw 3129	. 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 2111  ∀wral 3047  ∃wrex 3056  ∪ cuni 4856  ^◡ccnv 5613  ran crn 5615   “ cima 5617  Fun wfun 6475  (class class class)co 7346   ↑_m cmap 8750  sigAlgebracsiga 34121  MblFnMcmbfm 34262

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-mbfm 34263

This theorem is referenced by:  orvcval4  34474