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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elmthm | Structured version Visualization version GIF version |
Description: A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mthmval.r | ⊢ 𝑅 = (mStRed‘𝑇) |
mthmval.j | ⊢ 𝐽 = (mPPSt‘𝑇) |
mthmval.u | ⊢ 𝑈 = (mThm‘𝑇) |
Ref | Expression |
---|---|
elmthm | ⊢ (𝑋 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mthmval.r | . . . 4 ⊢ 𝑅 = (mStRed‘𝑇) | |
2 | mthmval.j | . . . 4 ⊢ 𝐽 = (mPPSt‘𝑇) | |
3 | mthmval.u | . . . 4 ⊢ 𝑈 = (mThm‘𝑇) | |
4 | 1, 2, 3 | mthmval 32935 | . . 3 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽)) |
5 | 4 | eleq2i 2881 | . 2 ⊢ (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡𝑅 “ (𝑅 “ 𝐽))) |
6 | eqid 2798 | . . . . 5 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇) | |
7 | 6, 1 | msrf 32902 | . . . 4 ⊢ 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) |
8 | ffn 6487 | . . . 4 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇)) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ 𝑅 Fn (mPreSt‘𝑇) |
10 | elpreima 6805 | . . 3 ⊢ (𝑅 Fn (mPreSt‘𝑇) → (𝑋 ∈ (◡𝑅 “ (𝑅 “ 𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽)))) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ (𝑋 ∈ (◡𝑅 “ (𝑅 “ 𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽))) |
12 | 6, 2 | mppspst 32934 | . . . . 5 ⊢ 𝐽 ⊆ (mPreSt‘𝑇) |
13 | fvelimab 6712 | . . . . 5 ⊢ ((𝑅 Fn (mPreSt‘𝑇) ∧ 𝐽 ⊆ (mPreSt‘𝑇)) → ((𝑅‘𝑋) ∈ (𝑅 “ 𝐽) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))) | |
14 | 9, 12, 13 | mp2an 691 | . . . 4 ⊢ ((𝑅‘𝑋) ∈ (𝑅 “ 𝐽) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)) |
15 | 14 | anbi2i 625 | . . 3 ⊢ ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))) |
16 | 12 | sseli 3911 | . . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ∈ (mPreSt‘𝑇)) |
17 | 6, 1 | msrrcl 32903 | . . . . . 6 ⊢ ((𝑅‘𝑥) = (𝑅‘𝑋) → (𝑥 ∈ (mPreSt‘𝑇) ↔ 𝑋 ∈ (mPreSt‘𝑇))) |
18 | 16, 17 | syl5ibcom 248 | . . . . 5 ⊢ (𝑥 ∈ 𝐽 → ((𝑅‘𝑥) = (𝑅‘𝑋) → 𝑋 ∈ (mPreSt‘𝑇))) |
19 | 18 | rexlimiv 3239 | . . . 4 ⊢ (∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋) → 𝑋 ∈ (mPreSt‘𝑇)) |
20 | 19 | pm4.71ri 564 | . . 3 ⊢ (∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))) |
21 | 15, 20 | bitr4i 281 | . 2 ⊢ ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽)) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)) |
22 | 5, 11, 21 | 3bitri 300 | 1 ⊢ (𝑋 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ⊆ wss 3881 ◡ccnv 5518 “ cima 5522 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 mPreStcmpst 32833 mStRedcmsr 32834 mPPStcmpps 32838 mThmcmthm 32839 |
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-ov 7138 df-oprab 7139 df-1st 7671 df-2nd 7672 df-mpst 32853 df-msr 32854 df-mpps 32858 df-mthm 32859 |
This theorem is referenced by: mthmi 32937 mthmpps 32942 |
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