Theorem mthmi 35032

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 6900	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑅‘𝑥) = (𝑅‘𝑌) ↔ (𝑅‘𝑋) = (𝑅‘𝑌)))
2	1	rspcev 3612	. 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 35031	. 2 ⊢ (𝑌 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑌))
7	2, 6	sylibr 233	1 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑅‘𝑋) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∃wrex 3069  ‘cfv 6543  mStRedcmsr 34929  mPPStcmpps 34933  mThmcmthm 34934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-1st 7979  df-2nd 7980  df-mpst 34948  df-msr 34949  df-mpps 34953  df-mthm 34954

This theorem is referenced by:  mppsthm  35034  mthmblem  35035  mthmpps  35037