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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 33964 | . . . 4 ⊢ 𝑅:𝑃⟶𝑃 |
4 | ffn 6665 | . . . 4 ⊢ (𝑅:𝑃⟶𝑃 → 𝑅 Fn 𝑃) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 𝑅 Fn 𝑃 |
6 | dffn4 6759 | . . 3 ⊢ (𝑅 Fn 𝑃 ↔ 𝑅:𝑃–onto→ran 𝑅) | |
7 | 5, 6 | mpbi 229 | . 2 ⊢ 𝑅:𝑃–onto→ran 𝑅 |
8 | mstaval.s | . . . 4 ⊢ 𝑆 = (mStat‘𝑇) | |
9 | 2, 8 | mstaval 33966 | . . 3 ⊢ 𝑆 = ran 𝑅 |
10 | foeq3 6751 | . . 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 230 | 1 ⊢ 𝑅:𝑃–onto→𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ran crn 5632 Fn wfn 6488 ⟶wf 6489 –onto→wfo 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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