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Theorem msrfo 31204
Description: The reduct of a pre-statement is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mstaval.r 𝑅 = (mStRed‘𝑇)
mstaval.s 𝑆 = (mStat‘𝑇)
msrfo.p 𝑃 = (mPreSt‘𝑇)
Assertion
Ref Expression
msrfo 𝑅:𝑃onto𝑆

Proof of Theorem msrfo
StepHypRef Expression
1 msrfo.p . . . . 5 𝑃 = (mPreSt‘𝑇)
2 mstaval.r . . . . 5 𝑅 = (mStRed‘𝑇)
31, 2msrf 31200 . . . 4 𝑅:𝑃𝑃
4 ffn 6012 . . . 4 (𝑅:𝑃𝑃𝑅 Fn 𝑃)
53, 4ax-mp 5 . . 3 𝑅 Fn 𝑃
6 dffn4 6088 . . 3 (𝑅 Fn 𝑃𝑅:𝑃onto→ran 𝑅)
75, 6mpbi 220 . 2 𝑅:𝑃onto→ran 𝑅
8 mstaval.s . . . 4 𝑆 = (mStat‘𝑇)
92, 8mstaval 31202 . . 3 𝑆 = ran 𝑅
10 foeq3 6080 . . 3 (𝑆 = ran 𝑅 → (𝑅:𝑃onto𝑆𝑅:𝑃onto→ran 𝑅))
119, 10ax-mp 5 . 2 (𝑅:𝑃onto𝑆𝑅:𝑃onto→ran 𝑅)
127, 11mpbir 221 1 𝑅:𝑃onto𝑆
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  ran crn 5085   Fn wfn 5852  wf 5853  ontowfo 5855  cfv 5857  mPreStcmpst 31131  mStRedcmsr 31132  mStatcmsta 31133
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-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  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 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-ot 4164  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-1st 7128  df-2nd 7129  df-mpst 31151  df-msr 31152  df-msta 31153
This theorem is referenced by: (None)
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