Theorem mthmblem 36093

Description: Lemma for mthmb 36094. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mthmb.r	⊢ 𝑅 = (mStRed‘𝑇)
mthmb.u	⊢ 𝑈 = (mThm‘𝑇)

Assertion

Ref	Expression
mthmblem	⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 → 𝑌 ∈ 𝑈))

Proof of Theorem mthmblem

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mthmb.r	. . . . 5 ⊢ 𝑅 = (mStRed‘𝑇)
2		eqid 2766	. . . . 5 ⊢ (mPPSt‘𝑇) = (mPPSt‘𝑇)
3		mthmb.u	. . . . 5 ⊢ 𝑈 = (mThm‘𝑇)
4	1, 2, 3	mthmval 36088	. . . 4 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ (mPPSt‘𝑇)))
5	4	eleq2i 2858	. . 3 ⊢ (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡𝑅 “ (𝑅 “ (mPPSt‘𝑇))))
6		eqid 2766	. . . . . 6 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
7	6, 1	msrf 36055	. . . . 5 ⊢ 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
8		ffn 6712	. . . . 5 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
9	7, 8	ax-mp 5	. . . 4 ⊢ 𝑅 Fn (mPreSt‘𝑇)
10		elpreima 7060	. . . 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 278	. 2 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))))
13		eleq1 2854	. . . 4 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) ↔ (𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))))
14		ffun 6715	. . . . . . 7 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → Fun 𝑅)
15	7, 14	ax-mp 5	. . . . . 6 ⊢ Fun 𝑅
16		fvelima 6953	. . . . . 6 ⊢ ((Fun 𝑅 ∧ (𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅‘𝑥) = (𝑅‘𝑌))
17	15, 16	mpan 703	. . . . 5 ⊢ ((𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅‘𝑥) = (𝑅‘𝑌))
18	1, 2, 3	mthmi 36090	. . . . . 6 ⊢ ((𝑥 ∈ (mPPSt‘𝑇) ∧ (𝑅‘𝑥) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈)
19	18	rexlimiva 3161	. . . . 5 ⊢ (∃𝑥 ∈ (mPPSt‘𝑇)(𝑅‘𝑥) = (𝑅‘𝑌) → 𝑌 ∈ 𝑈)
20	17, 19	syl 18	. . . 4 ⊢ ((𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → 𝑌 ∈ 𝑈)
21	13, 20	biimtrdi 256	. . 3 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) → 𝑌 ∈ 𝑈))
22	21	adantld 496	. 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))) → 𝑌 ∈ 𝑈))
23	12, 22	biimtrid 245	1 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 → 𝑌 ∈ 𝑈))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3092  ^◡ccnv 5665   “ cima 5669  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  mPreStcmpst 35986  mStRedcmsr 35987  mPPStcmpps 35991  mThmcmthm 35992

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-ot 4603  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-1st 7995  df-2nd 7996  df-mpst 36006  df-msr 36007  df-mpps 36011  df-mthm 36012

This theorem is used by:  mthmb  36094