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Theorem cndprobval 33432
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 7412 . 2 (𝐴(cprobβ€˜π‘ƒ)𝐡) = ((cprobβ€˜π‘ƒ)β€˜βŸ¨π΄, 𝐡⟩)
2 df-cndprob 33431 . . . . 5 cprob = (𝑝 ∈ Prob ↦ (π‘Ž ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((π‘β€˜(π‘Ž ∩ 𝑏)) / (π‘β€˜π‘))))
3 dmeq 5904 . . . . . 6 (𝑝 = 𝑃 β†’ dom 𝑝 = dom 𝑃)
4 fveq1 6891 . . . . . . 7 (𝑝 = 𝑃 β†’ (π‘β€˜(π‘Ž ∩ 𝑏)) = (π‘ƒβ€˜(π‘Ž ∩ 𝑏)))
5 fveq1 6891 . . . . . . 7 (𝑝 = 𝑃 β†’ (π‘β€˜π‘) = (π‘ƒβ€˜π‘))
64, 5oveq12d 7427 . . . . . 6 (𝑝 = 𝑃 β†’ ((π‘β€˜(π‘Ž ∩ 𝑏)) / (π‘β€˜π‘)) = ((π‘ƒβ€˜(π‘Ž ∩ 𝑏)) / (π‘ƒβ€˜π‘)))
73, 3, 6mpoeq123dv 7484 . . . . 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 7894 . . . . . 6 (𝑃 ∈ Prob β†’ dom 𝑃 ∈ V)
10 mpoexga 8064 . . . . . 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 585 . . . . 5 (𝑃 ∈ Prob β†’ (π‘Ž ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((π‘ƒβ€˜(π‘Ž ∩ 𝑏)) / (π‘ƒβ€˜π‘))) ∈ V)
122, 7, 8, 11fvmptd3 7022 . . . 4 (𝑃 ∈ Prob β†’ (cprobβ€˜π‘ƒ) = (π‘Ž ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((π‘ƒβ€˜(π‘Ž ∩ 𝑏)) / (π‘ƒβ€˜π‘))))
13123ad2ant1 1134 . . 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 770 . . . . . 6 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ (π‘Ž = 𝐴 ∧ 𝑏 = 𝐡)) β†’ π‘Ž = 𝐴)
15 simprr 772 . . . . . 6 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ (π‘Ž = 𝐴 ∧ 𝑏 = 𝐡)) β†’ 𝑏 = 𝐡)
1614, 15ineq12d 4214 . . . . 5 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ (π‘Ž = 𝐴 ∧ 𝑏 = 𝐡)) β†’ (π‘Ž ∩ 𝑏) = (𝐴 ∩ 𝐡))
1716fveq2d 6896 . . . 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 6896 . . . 4 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ (π‘Ž = 𝐴 ∧ 𝑏 = 𝐡)) β†’ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜π΅))
1917, 18oveq12d 7427 . . 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 1138 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ 𝐴 ∈ dom 𝑃)
21 simp3 1139 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ 𝐡 ∈ dom 𝑃)
22 ovexd 7444 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ ((π‘ƒβ€˜(𝐴 ∩ 𝐡)) / (π‘ƒβ€˜π΅)) ∈ V)
2313, 19, 20, 21, 22ovmpod 7560 . 2 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ (𝐴(cprobβ€˜π‘ƒ)𝐡) = ((π‘ƒβ€˜(𝐴 ∩ 𝐡)) / (π‘ƒβ€˜π΅)))
241, 23eqtr3id 2787 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∩ cin 3948  βŸ¨cop 4635  dom cdm 5677  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411   / cdiv 11871  Probcprb 33406  cprobccprob 33430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-cndprob 33431
This theorem is referenced by:  cndprobin  33433  cndprob01  33434  cndprobtot  33435  cndprobnul  33436  cndprobprob  33437
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