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Theorem isanmbfm 30123
Description: The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypotheses
Ref Expression
mbfmf.1 (𝜑𝑆 ran sigAlgebra)
mbfmf.2 (𝜑𝑇 ran sigAlgebra)
mbfmf.3 (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
Assertion
Ref Expression
isanmbfm (𝜑𝐹 ran MblFnM)

Proof of Theorem isanmbfm
Dummy variables 𝑡 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mbfmf.1 . . 3 (𝜑𝑆 ran sigAlgebra)
2 mbfmf.2 . . 3 (𝜑𝑇 ran sigAlgebra)
3 mbfmf.3 . . . 4 (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
41, 2ismbfm 30119 . . . 4 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
53, 4mpbid 222 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
6 unieq 4415 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
76oveq2d 6626 . . . . . 6 (𝑠 = 𝑆 → ( 𝑡𝑚 𝑠) = ( 𝑡𝑚 𝑆))
87eleq2d 2684 . . . . 5 (𝑠 = 𝑆 → (𝐹 ∈ ( 𝑡𝑚 𝑠) ↔ 𝐹 ∈ ( 𝑡𝑚 𝑆)))
9 eleq2 2687 . . . . . 6 (𝑠 = 𝑆 → ((𝐹𝑥) ∈ 𝑠 ↔ (𝐹𝑥) ∈ 𝑆))
109ralbidv 2981 . . . . 5 (𝑠 = 𝑆 → (∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠 ↔ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑆))
118, 10anbi12d 746 . . . 4 (𝑠 = 𝑆 → ((𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠) ↔ (𝐹 ∈ ( 𝑡𝑚 𝑆) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑆)))
12 unieq 4415 . . . . . . 7 (𝑡 = 𝑇 𝑡 = 𝑇)
1312oveq1d 6625 . . . . . 6 (𝑡 = 𝑇 → ( 𝑡𝑚 𝑆) = ( 𝑇𝑚 𝑆))
1413eleq2d 2684 . . . . 5 (𝑡 = 𝑇 → (𝐹 ∈ ( 𝑡𝑚 𝑆) ↔ 𝐹 ∈ ( 𝑇𝑚 𝑆)))
15 raleq 3130 . . . . 5 (𝑡 = 𝑇 → (∀𝑥𝑡 (𝐹𝑥) ∈ 𝑆 ↔ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
1614, 15anbi12d 746 . . . 4 (𝑡 = 𝑇 → ((𝐹 ∈ ( 𝑡𝑚 𝑆) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑆) ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
1711, 16rspc2ev 3312 . . 3 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra ∧ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)) → ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
181, 2, 5, 17syl3anc 1323 . 2 (𝜑 → ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
19 elunirnmbfm 30120 . 2 (𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
2018, 19sylibr 224 1 (𝜑𝐹 ran MblFnM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908   cuni 4407  ccnv 5078  ran crn 5080  cima 5082  (class class class)co 6610  𝑚 cmap 7809  sigAlgebracsiga 29975  MblFnMcmbfm 30117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-mbfm 30118
This theorem is referenced by:  mbfmbfm  30125  orvcval4  30327
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