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Theorem msrfo 33968
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 33964 . . . 4 𝑅:𝑃𝑃
4 ffn 6665 . . . 4 (𝑅:𝑃𝑃𝑅 Fn 𝑃)
53, 4ax-mp 5 . . 3 𝑅 Fn 𝑃
6 dffn4 6759 . . 3 (𝑅 Fn 𝑃𝑅:𝑃onto→ran 𝑅)
75, 6mpbi 229 . 2 𝑅:𝑃onto→ran 𝑅
8 mstaval.s . . . 4 𝑆 = (mStat‘𝑇)
92, 8mstaval 33966 . . 3 𝑆 = ran 𝑅
10 foeq3 6751 . . 3 (𝑆 = ran 𝑅 → (𝑅:𝑃onto𝑆𝑅:𝑃onto→ran 𝑅))
119, 10ax-mp 5 . 2 (𝑅:𝑃onto𝑆𝑅:𝑃onto→ran 𝑅)
127, 11mpbir 230 1 𝑅:𝑃onto𝑆
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  ran crn 5632   Fn wfn 6488  wf 6489  ontowfo 6491  cfv 6493  mPreStcmpst 33895  mStRedcmsr 33896  mStatcmsta 33897
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 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-1st 7913  df-2nd 7914  df-mpst 33915  df-msr 33916  df-msta 33917
This theorem is referenced by: (None)
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