Theorem mbfmfun 30857

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 30856	. . 3 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠))
3	2	biimpi 208	. 2 ⊢ (𝐹 ∈ ∪ ran MblFnM → ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠))
4		elmapfun 8151	. . . . 5 ⊢ (𝐹 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) → Fun 𝐹)
5	4	adantr 474	. . . 4 ⊢ ((𝐹 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠) → Fun 𝐹)
6	5	rexlimivw 3238	. . 3 ⊢ (∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠) → Fun 𝐹)
7	6	rexlimivw 3238	. 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 386   ∈ wcel 2164  ∀wral 3117  ∃wrex 3118  ∪ cuni 4660  ^◡ccnv 5345  ran crn 5347   “ cima 5349  Fun wfun 6121  (class class class)co 6910   ↑_𝑚 cmap 8127  sigAlgebracsiga 30711  MblFnMcmbfm 30853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-map 8129  df-mbfm 30854

This theorem is referenced by:  orvcval4  31064