Theorem mthmblem 35774

Description: Lemma for mthmb 35775. (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 2736	. . . . 5 ⊢ (mPPSt‘𝑇) = (mPPSt‘𝑇)
3		mthmb.u	. . . . 5 ⊢ 𝑈 = (mThm‘𝑇)
4	1, 2, 3	mthmval 35769	. . . 4 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ (mPPSt‘𝑇)))
5	4	eleq2i 2828	. . 3 ⊢ (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡𝑅 “ (𝑅 “ (mPPSt‘𝑇))))
6		eqid 2736	. . . . . 6 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
7	6, 1	msrf 35736	. . . . 5 ⊢ 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
8		ffn 6662	. . . . 5 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
9	7, 8	ax-mp 5	. . . 4 ⊢ 𝑅 Fn (mPreSt‘𝑇)
10		elpreima 7003	. . . 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 275	. 2 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))))
13		eleq1 2824	. . . 4 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) ↔ (𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))))
14		ffun 6665	. . . . . . 7 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → Fun 𝑅)
15	7, 14	ax-mp 5	. . . . . 6 ⊢ Fun 𝑅
16		fvelima 6899	. . . . . 6 ⊢ ((Fun 𝑅 ∧ (𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅‘𝑥) = (𝑅‘𝑌))
17	15, 16	mpan 690	. . . . 5 ⊢ ((𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅‘𝑥) = (𝑅‘𝑌))
18	1, 2, 3	mthmi 35771	. . . . . 6 ⊢ ((𝑥 ∈ (mPPSt‘𝑇) ∧ (𝑅‘𝑥) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈)
19	18	rexlimiva 3129	. . . . 5 ⊢ (∃𝑥 ∈ (mPPSt‘𝑇)(𝑅‘𝑥) = (𝑅‘𝑌) → 𝑌 ∈ 𝑈)
20	17, 19	syl 17	. . . 4 ⊢ ((𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → 𝑌 ∈ 𝑈)
21	13, 20	biimtrdi 253	. . 3 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) → 𝑌 ∈ 𝑈))
22	21	adantld 490	. 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))) → 𝑌 ∈ 𝑈))
23	12, 22	biimtrid 242	1 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 → 𝑌 ∈ 𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  ^◡ccnv 5623   “ cima 5627  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  mPreStcmpst 35667  mStRedcmsr 35668  mPPStcmpps 35672  mThmcmthm 35673

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-ot 4589  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-1st 7933  df-2nd 7934  df-mpst 35687  df-msr 35688  df-mpps 35692  df-mthm 35693

This theorem is referenced by:  mthmb  35775