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| Description: The reduct of a pre-statement is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.) | 
| Ref | Expression | 
|---|---|
| mstaval.r | ⊢ 𝑅 = (mStRed‘𝑇) | 
| mstaval.s | ⊢ 𝑆 = (mStat‘𝑇) | 
| msrfo.p | ⊢ 𝑃 = (mPreSt‘𝑇) | 
| Ref | Expression | 
|---|---|
| msrfo | ⊢ 𝑅:𝑃–onto→𝑆 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | msrfo.p | . . . . 5 ⊢ 𝑃 = (mPreSt‘𝑇) | |
| 2 | mstaval.r | . . . . 5 ⊢ 𝑅 = (mStRed‘𝑇) | |
| 3 | 1, 2 | msrf 35547 | . . . 4 ⊢ 𝑅:𝑃⟶𝑃 | 
| 4 | ffn 6736 | . . . 4 ⊢ (𝑅:𝑃⟶𝑃 → 𝑅 Fn 𝑃) | |
| 5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 𝑅 Fn 𝑃 | 
| 6 | dffn4 6826 | . . 3 ⊢ (𝑅 Fn 𝑃 ↔ 𝑅:𝑃–onto→ran 𝑅) | |
| 7 | 5, 6 | mpbi 230 | . 2 ⊢ 𝑅:𝑃–onto→ran 𝑅 | 
| 8 | mstaval.s | . . . 4 ⊢ 𝑆 = (mStat‘𝑇) | |
| 9 | 2, 8 | mstaval 35549 | . . 3 ⊢ 𝑆 = ran 𝑅 | 
| 10 | foeq3 6818 | . . 3 ⊢ (𝑆 = ran 𝑅 → (𝑅:𝑃–onto→𝑆 ↔ 𝑅:𝑃–onto→ran 𝑅)) | |
| 11 | 9, 10 | ax-mp 5 | . 2 ⊢ (𝑅:𝑃–onto→𝑆 ↔ 𝑅:𝑃–onto→ran 𝑅) | 
| 12 | 7, 11 | mpbir 231 | 1 ⊢ 𝑅:𝑃–onto→𝑆 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ran crn 5686 Fn wfn 6556 ⟶wf 6557 –onto→wfo 6559 ‘cfv 6561 mPreStcmpst 35478 mStRedcmsr 35479 mStatcmsta 35480 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1st 8014 df-2nd 8015 df-mpst 35498 df-msr 35499 df-msta 35500 | 
| This theorem is referenced by: (None) | 
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