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Theorem msrfo 32853
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 32849 . . . 4 𝑅:𝑃𝑃
4 ffn 6504 . . . 4 (𝑅:𝑃𝑃𝑅 Fn 𝑃)
53, 4ax-mp 5 . . 3 𝑅 Fn 𝑃
6 dffn4 6588 . . 3 (𝑅 Fn 𝑃𝑅:𝑃onto→ran 𝑅)
75, 6mpbi 233 . 2 𝑅:𝑃onto→ran 𝑅
8 mstaval.s . . . 4 𝑆 = (mStat‘𝑇)
92, 8mstaval 32851 . . 3 𝑆 = ran 𝑅
10 foeq3 6580 . . 3 (𝑆 = ran 𝑅 → (𝑅:𝑃onto𝑆𝑅:𝑃onto→ran 𝑅))
119, 10ax-mp 5 . 2 (𝑅:𝑃onto𝑆𝑅:𝑃onto→ran 𝑅)
127, 11mpbir 234 1 𝑅:𝑃onto𝑆
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  ran crn 5544   Fn wfn 6339  wf 6340  ontowfo 6342  cfv 6344  mPreStcmpst 32780  mStRedcmsr 32781  mStatcmsta 32782
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-ot 4560  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-1st 7685  df-2nd 7686  df-mpst 32800  df-msr 32801  df-msta 32802
This theorem is referenced by: (None)
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