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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 32789 | . . . 4 ⊢ 𝑅:𝑃⟶𝑃 |
4 | ffn 6514 | . . . 4 ⊢ (𝑅:𝑃⟶𝑃 → 𝑅 Fn 𝑃) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 𝑅 Fn 𝑃 |
6 | dffn4 6596 | . . 3 ⊢ (𝑅 Fn 𝑃 ↔ 𝑅:𝑃–onto→ran 𝑅) | |
7 | 5, 6 | mpbi 232 | . 2 ⊢ 𝑅:𝑃–onto→ran 𝑅 |
8 | mstaval.s | . . . 4 ⊢ 𝑆 = (mStat‘𝑇) | |
9 | 2, 8 | mstaval 32791 | . . 3 ⊢ 𝑆 = ran 𝑅 |
10 | foeq3 6588 | . . 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 233 | 1 ⊢ 𝑅:𝑃–onto→𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ran crn 5556 Fn wfn 6350 ⟶wf 6351 –onto→wfo 6353 ‘cfv 6355 mPreStcmpst 32720 mStRedcmsr 32721 mStatcmsta 32722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-ot 4576 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-1st 7689 df-2nd 7690 df-mpst 32740 df-msr 32741 df-msta 32742 |
This theorem is referenced by: (None) |
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