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Mirrors > Home > MPE Home > Th. List > Mathboxes > mthmblem | Structured version Visualization version GIF version |
Description: Lemma for mthmb 32941. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mthmb.r | ⊢ 𝑅 = (mStRed‘𝑇) |
mthmb.u | ⊢ 𝑈 = (mThm‘𝑇) |
Ref | Expression |
---|---|
mthmblem | ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 → 𝑌 ∈ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mthmb.r | . . . . 5 ⊢ 𝑅 = (mStRed‘𝑇) | |
2 | eqid 2798 | . . . . 5 ⊢ (mPPSt‘𝑇) = (mPPSt‘𝑇) | |
3 | mthmb.u | . . . . 5 ⊢ 𝑈 = (mThm‘𝑇) | |
4 | 1, 2, 3 | mthmval 32935 | . . . 4 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ (mPPSt‘𝑇))) |
5 | 4 | eleq2i 2881 | . . 3 ⊢ (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡𝑅 “ (𝑅 “ (mPPSt‘𝑇)))) |
6 | eqid 2798 | . . . . . 6 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇) | |
7 | 6, 1 | msrf 32902 | . . . . 5 ⊢ 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) |
8 | ffn 6487 | . . . . 5 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇)) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ 𝑅 Fn (mPreSt‘𝑇) |
10 | elpreima 6805 | . . . 4 ⊢ (𝑅 Fn (mPreSt‘𝑇) → (𝑋 ∈ (◡𝑅 “ (𝑅 “ (mPPSt‘𝑇))) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))))) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ (𝑋 ∈ (◡𝑅 “ (𝑅 “ (mPPSt‘𝑇))) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)))) |
12 | 5, 11 | bitri 278 | . 2 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)))) |
13 | eleq1 2877 | . . . 4 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) ↔ (𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)))) | |
14 | ffun 6490 | . . . . . . 7 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → Fun 𝑅) | |
15 | 7, 14 | ax-mp 5 | . . . . . 6 ⊢ Fun 𝑅 |
16 | fvelima 6706 | . . . . . 6 ⊢ ((Fun 𝑅 ∧ (𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅‘𝑥) = (𝑅‘𝑌)) | |
17 | 15, 16 | mpan 689 | . . . . 5 ⊢ ((𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅‘𝑥) = (𝑅‘𝑌)) |
18 | 1, 2, 3 | mthmi 32937 | . . . . . 6 ⊢ ((𝑥 ∈ (mPPSt‘𝑇) ∧ (𝑅‘𝑥) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈) |
19 | 18 | rexlimiva 3240 | . . . . 5 ⊢ (∃𝑥 ∈ (mPPSt‘𝑇)(𝑅‘𝑥) = (𝑅‘𝑌) → 𝑌 ∈ 𝑈) |
20 | 17, 19 | syl 17 | . . . 4 ⊢ ((𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → 𝑌 ∈ 𝑈) |
21 | 13, 20 | syl6bi 256 | . . 3 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) → 𝑌 ∈ 𝑈)) |
22 | 21 | adantld 494 | . 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))) → 𝑌 ∈ 𝑈)) |
23 | 12, 22 | syl5bi 245 | 1 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 → 𝑌 ∈ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ◡ccnv 5518 “ cima 5522 Fun wfun 6318 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: mthmb 32941 |
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