Theorem mthmi 35582

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 6915	. . 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 35581	. 2 ⊢ (𝑌 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑌))
7	2, 6	sylibr 234	1 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑅‘𝑋) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  ‘cfv 6561  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:  mppsthm  35584  mthmblem  35585  mthmpps  35587