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Theorem isanmbfm 31622
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 31618 . . . 4 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
53, 4mpbid 235 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
6 unieq 4814 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
76oveq2d 7155 . . . . . 6 (𝑠 = 𝑆 → ( 𝑡m 𝑠) = ( 𝑡m 𝑆))
87eleq2d 2878 . . . . 5 (𝑠 = 𝑆 → (𝐹 ∈ ( 𝑡m 𝑠) ↔ 𝐹 ∈ ( 𝑡m 𝑆)))
9 eleq2 2881 . . . . . 6 (𝑠 = 𝑆 → ((𝐹𝑥) ∈ 𝑠 ↔ (𝐹𝑥) ∈ 𝑆))
109ralbidv 3165 . . . . 5 (𝑠 = 𝑆 → (∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠 ↔ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑆))
118, 10anbi12d 633 . . . 4 (𝑠 = 𝑆 → ((𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠) ↔ (𝐹 ∈ ( 𝑡m 𝑆) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑆)))
12 unieq 4814 . . . . . . 7 (𝑡 = 𝑇 𝑡 = 𝑇)
1312oveq1d 7154 . . . . . 6 (𝑡 = 𝑇 → ( 𝑡m 𝑆) = ( 𝑇m 𝑆))
1413eleq2d 2878 . . . . 5 (𝑡 = 𝑇 → (𝐹 ∈ ( 𝑡m 𝑆) ↔ 𝐹 ∈ ( 𝑇m 𝑆)))
15 raleq 3361 . . . . 5 (𝑡 = 𝑇 → (∀𝑥𝑡 (𝐹𝑥) ∈ 𝑆 ↔ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
1614, 15anbi12d 633 . . . 4 (𝑡 = 𝑇 → ((𝐹 ∈ ( 𝑡m 𝑆) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑆) ↔ (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
1711, 16rspc2ev 3586 . . 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 1368 . 2 (𝜑 → ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
19 elunirnmbfm 31619 . 2 (𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
2018, 19sylibr 237 1 (𝜑𝐹 ran MblFnM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  wrex 3110   cuni 4803  ccnv 5522  ran crn 5524  cima 5526  (class class class)co 7139  m cmap 8393  sigAlgebracsiga 31475  MblFnMcmbfm 31616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-mbfm 31617
This theorem is referenced by:  mbfmbfm  31624  orvcval4  31826
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