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Theorem msrfo 31750
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 31746 . . . 4 𝑅:𝑃𝑃
4 ffn 6206 . . . 4 (𝑅:𝑃𝑃𝑅 Fn 𝑃)
53, 4ax-mp 5 . . 3 𝑅 Fn 𝑃
6 dffn4 6282 . . 3 (𝑅 Fn 𝑃𝑅:𝑃onto→ran 𝑅)
75, 6mpbi 220 . 2 𝑅:𝑃onto→ran 𝑅
8 mstaval.s . . . 4 𝑆 = (mStat‘𝑇)
92, 8mstaval 31748 . . 3 𝑆 = ran 𝑅
10 foeq3 6274 . . 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 1632  ran crn 5267   Fn wfn 6044  wf 6045  ontowfo 6047  cfv 6049  mPreStcmpst 31677  mStRedcmsr 31678  mStatcmsta 31679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-ot 4330  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-1st 7333  df-2nd 7334  df-mpst 31697  df-msr 31698  df-msta 31699
This theorem is referenced by: (None)
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