Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > mthmi | Structured version Visualization version GIF version |
Description: A statement whose reduct is the reduct of a provable pre-statement is a theorem. (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 |
---|---|
mthmi | ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑅‘𝑋) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveqeq2 6704 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑅‘𝑥) = (𝑅‘𝑌) ↔ (𝑅‘𝑋) = (𝑅‘𝑌))) | |
2 | 1 | rspcev 3527 | . 2 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑅‘𝑋) = (𝑅‘𝑌)) → ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑌)) |
3 | mthmval.r | . . 3 ⊢ 𝑅 = (mStRed‘𝑇) | |
4 | mthmval.j | . . 3 ⊢ 𝐽 = (mPPSt‘𝑇) | |
5 | mthmval.u | . . 3 ⊢ 𝑈 = (mThm‘𝑇) | |
6 | 3, 4, 5 | elmthm 33205 | . 2 ⊢ (𝑌 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑌)) |
7 | 2, 6 | sylibr 237 | 1 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑅‘𝑋) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 ‘cfv 6358 mStRedcmsr 33103 mPPStcmpps 33107 mThmcmthm 33108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-ot 4536 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-1st 7739 df-2nd 7740 df-mpst 33122 df-msr 33123 df-mpps 33127 df-mthm 33128 |
This theorem is referenced by: mppsthm 33208 mthmblem 33209 mthmpps 33211 |
Copyright terms: Public domain | W3C validator |