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Theorem cndprobval 33501
Description: The value of the conditional probability , i.e. the probability for the event 𝐴, given 𝐡, under the probability law 𝑃. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Assertion
Ref Expression
cndprobval ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ ((cprobβ€˜π‘ƒ)β€˜βŸ¨π΄, 𝐡⟩) = ((π‘ƒβ€˜(𝐴 ∩ 𝐡)) / (π‘ƒβ€˜π΅)))

Proof of Theorem cndprobval
Dummy variables π‘Ž 𝑏 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 7414 . 2 (𝐴(cprobβ€˜π‘ƒ)𝐡) = ((cprobβ€˜π‘ƒ)β€˜βŸ¨π΄, 𝐡⟩)
2 df-cndprob 33500 . . . . 5 cprob = (𝑝 ∈ Prob ↦ (π‘Ž ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((π‘β€˜(π‘Ž ∩ 𝑏)) / (π‘β€˜π‘))))
3 dmeq 5903 . . . . . 6 (𝑝 = 𝑃 β†’ dom 𝑝 = dom 𝑃)
4 fveq1 6890 . . . . . . 7 (𝑝 = 𝑃 β†’ (π‘β€˜(π‘Ž ∩ 𝑏)) = (π‘ƒβ€˜(π‘Ž ∩ 𝑏)))
5 fveq1 6890 . . . . . . 7 (𝑝 = 𝑃 β†’ (π‘β€˜π‘) = (π‘ƒβ€˜π‘))
64, 5oveq12d 7429 . . . . . 6 (𝑝 = 𝑃 β†’ ((π‘β€˜(π‘Ž ∩ 𝑏)) / (π‘β€˜π‘)) = ((π‘ƒβ€˜(π‘Ž ∩ 𝑏)) / (π‘ƒβ€˜π‘)))
73, 3, 6mpoeq123dv 7486 . . . . 5 (𝑝 = 𝑃 β†’ (π‘Ž ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((π‘β€˜(π‘Ž ∩ 𝑏)) / (π‘β€˜π‘))) = (π‘Ž ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((π‘ƒβ€˜(π‘Ž ∩ 𝑏)) / (π‘ƒβ€˜π‘))))
8 id 22 . . . . 5 (𝑃 ∈ Prob β†’ 𝑃 ∈ Prob)
9 dmexg 7896 . . . . . 6 (𝑃 ∈ Prob β†’ dom 𝑃 ∈ V)
10 mpoexga 8066 . . . . . 6 ((dom 𝑃 ∈ V ∧ dom 𝑃 ∈ V) β†’ (π‘Ž ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((π‘ƒβ€˜(π‘Ž ∩ 𝑏)) / (π‘ƒβ€˜π‘))) ∈ V)
119, 9, 10syl2anc 584 . . . . 5 (𝑃 ∈ Prob β†’ (π‘Ž ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((π‘ƒβ€˜(π‘Ž ∩ 𝑏)) / (π‘ƒβ€˜π‘))) ∈ V)
122, 7, 8, 11fvmptd3 7021 . . . 4 (𝑃 ∈ Prob β†’ (cprobβ€˜π‘ƒ) = (π‘Ž ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((π‘ƒβ€˜(π‘Ž ∩ 𝑏)) / (π‘ƒβ€˜π‘))))
13123ad2ant1 1133 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ (cprobβ€˜π‘ƒ) = (π‘Ž ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((π‘ƒβ€˜(π‘Ž ∩ 𝑏)) / (π‘ƒβ€˜π‘))))
14 simprl 769 . . . . . 6 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ (π‘Ž = 𝐴 ∧ 𝑏 = 𝐡)) β†’ π‘Ž = 𝐴)
15 simprr 771 . . . . . 6 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ (π‘Ž = 𝐴 ∧ 𝑏 = 𝐡)) β†’ 𝑏 = 𝐡)
1614, 15ineq12d 4213 . . . . 5 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ (π‘Ž = 𝐴 ∧ 𝑏 = 𝐡)) β†’ (π‘Ž ∩ 𝑏) = (𝐴 ∩ 𝐡))
1716fveq2d 6895 . . . 4 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ (π‘Ž = 𝐴 ∧ 𝑏 = 𝐡)) β†’ (π‘ƒβ€˜(π‘Ž ∩ 𝑏)) = (π‘ƒβ€˜(𝐴 ∩ 𝐡)))
1815fveq2d 6895 . . . 4 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ (π‘Ž = 𝐴 ∧ 𝑏 = 𝐡)) β†’ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜π΅))
1917, 18oveq12d 7429 . . 3 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ (π‘Ž = 𝐴 ∧ 𝑏 = 𝐡)) β†’ ((π‘ƒβ€˜(π‘Ž ∩ 𝑏)) / (π‘ƒβ€˜π‘)) = ((π‘ƒβ€˜(𝐴 ∩ 𝐡)) / (π‘ƒβ€˜π΅)))
20 simp2 1137 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ 𝐴 ∈ dom 𝑃)
21 simp3 1138 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ 𝐡 ∈ dom 𝑃)
22 ovexd 7446 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ ((π‘ƒβ€˜(𝐴 ∩ 𝐡)) / (π‘ƒβ€˜π΅)) ∈ V)
2313, 19, 20, 21, 22ovmpod 7562 . 2 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ (𝐴(cprobβ€˜π‘ƒ)𝐡) = ((π‘ƒβ€˜(𝐴 ∩ 𝐡)) / (π‘ƒβ€˜π΅)))
241, 23eqtr3id 2786 1 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ ((cprobβ€˜π‘ƒ)β€˜βŸ¨π΄, 𝐡⟩) = ((π‘ƒβ€˜(𝐴 ∩ 𝐡)) / (π‘ƒβ€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∩ cin 3947  βŸ¨cop 4634  dom cdm 5676  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413   / cdiv 11873  Probcprb 33475  cprobccprob 33499
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-cndprob 33500
This theorem is referenced by:  cndprobin  33502  cndprob01  33503  cndprobtot  33504  cndprobnul  33505  cndprobprob  33506
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