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Mirrors > Home > MPE Home > Th. List > Mathboxes > msrfo | Structured version Visualization version GIF version |
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 32902 | . . . 4 ⊢ 𝑅:𝑃⟶𝑃 |
4 | ffn 6487 | . . . 4 ⊢ (𝑅:𝑃⟶𝑃 → 𝑅 Fn 𝑃) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 𝑅 Fn 𝑃 |
6 | dffn4 6571 | . . 3 ⊢ (𝑅 Fn 𝑃 ↔ 𝑅:𝑃–onto→ran 𝑅) | |
7 | 5, 6 | mpbi 233 | . 2 ⊢ 𝑅:𝑃–onto→ran 𝑅 |
8 | mstaval.s | . . . 4 ⊢ 𝑆 = (mStat‘𝑇) | |
9 | 2, 8 | mstaval 32904 | . . 3 ⊢ 𝑆 = ran 𝑅 |
10 | foeq3 6563 | . . 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 234 | 1 ⊢ 𝑅:𝑃–onto→𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ran crn 5520 Fn wfn 6319 ⟶wf 6320 –onto→wfo 6322 ‘cfv 6324 mPreStcmpst 32833 mStRedcmsr 32834 mStatcmsta 32835 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
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 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-ot 4534 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-1st 7671 df-2nd 7672 df-mpst 32853 df-msr 32854 df-msta 32855 |
This theorem is referenced by: (None) |
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