Theorem mthmi 32824

Description: A statement whose reduct is the reduct of a provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mthmval.r	⊢ 𝑅 = (mStRed‘𝑇)
mthmval.j	⊢ 𝐽 = (mPPSt‘𝑇)
mthmval.u	⊢ 𝑈 = (mThm‘𝑇)

Assertion

Ref	Expression
mthmi	⊢ ((𝑋 ∈ 𝐽 ∧ (𝑅‘𝑋) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈)

Proof of Theorem mthmi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveqeq2 6678	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑅‘𝑥) = (𝑅‘𝑌) ↔ (𝑅‘𝑋) = (𝑅‘𝑌)))
2	1	rspcev 3622	. 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 32823	. 2 ⊢ (𝑌 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑌))
7	2, 6	sylibr 236	1 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑅‘𝑋) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  ‘cfv 6354  mStRedcmsr 32721  mPPStcmpps 32725  mThmcmthm 32726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-ot 4575  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-1st 7688  df-2nd 7689  df-mpst 32740  df-msr 32741  df-mpps 32745  df-mthm 32746

This theorem is referenced by:  mppsthm  32826  mthmblem  32827  mthmpps  32829