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Mirrors > Home > MPE Home > Th. List > Mathboxes > mthmblem | Structured version Visualization version GIF version |
Description: Lemma for mthmb 30732. (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 2610 | . . . . 5 ⊢ (mPPSt‘𝑇) = (mPPSt‘𝑇) | |
3 | mthmb.u | . . . . 5 ⊢ 𝑈 = (mThm‘𝑇) | |
4 | 1, 2, 3 | mthmval 30726 | . . . 4 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ (mPPSt‘𝑇))) |
5 | 4 | eleq2i 2680 | . . 3 ⊢ (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡𝑅 “ (𝑅 “ (mPPSt‘𝑇)))) |
6 | eqid 2610 | . . . . . 6 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇) | |
7 | 6, 1 | msrf 30693 | . . . . 5 ⊢ 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) |
8 | ffn 5958 | . . . . 5 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇)) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ 𝑅 Fn (mPreSt‘𝑇) |
10 | elpreima 6245 | . . . 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 263 | . 2 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)))) |
13 | eleq1 2676 | . . . 4 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) ↔ (𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)))) | |
14 | ffun 5961 | . . . . . . 7 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → Fun 𝑅) | |
15 | 7, 14 | ax-mp 5 | . . . . . 6 ⊢ Fun 𝑅 |
16 | fvelima 6158 | . . . . . 6 ⊢ ((Fun 𝑅 ∧ (𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅‘𝑥) = (𝑅‘𝑌)) | |
17 | 15, 16 | mpan 702 | . . . . 5 ⊢ ((𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅‘𝑥) = (𝑅‘𝑌)) |
18 | 1, 2, 3 | mthmi 30728 | . . . . . 6 ⊢ ((𝑥 ∈ (mPPSt‘𝑇) ∧ (𝑅‘𝑥) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈) |
19 | 18 | rexlimiva 3010 | . . . . 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 242 | . . 3 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) → 𝑌 ∈ 𝑈)) |
22 | 21 | adantld 482 | . 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))) → 𝑌 ∈ 𝑈)) |
23 | 12, 22 | syl5bi 231 | 1 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 → 𝑌 ∈ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ◡ccnv 5037 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 mPreStcmpst 30624 mStRedcmsr 30625 mPPStcmpps 30629 mThmcmthm 30630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-ot 4134 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-1st 7059 df-2nd 7060 df-mpst 30644 df-msr 30645 df-mpps 30649 df-mthm 30650 |
This theorem is referenced by: mthmb 30732 |
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