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Theorem mbfmfun 30625
Description: A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
Hypothesis
Ref Expression
mbfmfun.1 (𝜑𝐹 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.
StepHypRef Expression
1 mbfmfun.1 . 2 (𝜑𝐹 ran MblFnM)
2 elunirnmbfm 30624 . . 3 (𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
32biimpi 206 . 2 (𝐹 ran MblFnM → ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
4 elmapfun 8047 . . . . 5 (𝐹 ∈ ( 𝑡𝑚 𝑠) → Fun 𝐹)
54adantr 472 . . . 4 ((𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠) → Fun 𝐹)
65rexlimivw 3167 . . 3 (∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠) → Fun 𝐹)
76rexlimivw 3167 . 2 (∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠) → Fun 𝐹)
81, 3, 73syl 18 1 (𝜑 → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  wral 3050  wrex 3051   cuni 4588  ccnv 5265  ran crn 5267  cima 5269  Fun wfun 6043  (class class class)co 6813  𝑚 cmap 8023  sigAlgebracsiga 30479  MblFnMcmbfm 30621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-mbfm 30622
This theorem is referenced by:  orvcval4  30831
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