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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 35580 | . . 3 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽)) | 
| 5 | 4 | eleq2i 2833 | . 2 ⊢ (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡𝑅 “ (𝑅 “ 𝐽))) | 
| 6 | eqid 2737 | . . . . 5 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇) | |
| 7 | 6, 1 | msrf 35547 | . . . 4 ⊢ 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) | 
| 8 | ffn 6736 | . . . 4 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇)) | |
| 9 | 7, 8 | ax-mp 5 | . . 3 ⊢ 𝑅 Fn (mPreSt‘𝑇) | 
| 10 | elpreima 7078 | . . 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 35579 | . . . . 5 ⊢ 𝐽 ⊆ (mPreSt‘𝑇) | 
| 13 | fvelimab 6981 | . . . . 5 ⊢ ((𝑅 Fn (mPreSt‘𝑇) ∧ 𝐽 ⊆ (mPreSt‘𝑇)) → ((𝑅‘𝑋) ∈ (𝑅 “ 𝐽) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))) | |
| 14 | 9, 12, 13 | mp2an 692 | . . . 4 ⊢ ((𝑅‘𝑋) ∈ (𝑅 “ 𝐽) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)) | 
| 15 | 14 | anbi2i 623 | . . 3 ⊢ ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))) | 
| 16 | 12 | sseli 3979 | . . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ∈ (mPreSt‘𝑇)) | 
| 17 | 6, 1 | msrrcl 35548 | . . . . . 6 ⊢ ((𝑅‘𝑥) = (𝑅‘𝑋) → (𝑥 ∈ (mPreSt‘𝑇) ↔ 𝑋 ∈ (mPreSt‘𝑇))) | 
| 18 | 16, 17 | syl5ibcom 245 | . . . . 5 ⊢ (𝑥 ∈ 𝐽 → ((𝑅‘𝑥) = (𝑅‘𝑋) → 𝑋 ∈ (mPreSt‘𝑇))) | 
| 19 | 18 | rexlimiv 3148 | . . . 4 ⊢ (∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋) → 𝑋 ∈ (mPreSt‘𝑇)) | 
| 20 | 19 | pm4.71ri 560 | . . 3 ⊢ (∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))) | 
| 21 | 15, 20 | bitr4i 278 | . 2 ⊢ ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽)) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)) | 
| 22 | 5, 11, 21 | 3bitri 297 | 1 ⊢ (𝑋 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 ◡ccnv 5684 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 mPreStcmpst 35478 mStRedcmsr 35479 mPPStcmpps 35483 mThmcmthm 35484 | 
| 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-ov 7434 df-oprab 7435 df-1st 8014 df-2nd 8015 df-mpst 35498 df-msr 35499 df-mpps 35503 df-mthm 35504 | 
| This theorem is referenced by: mthmi 35582 mthmpps 35587 | 
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