Theorem mthmblem 35612

Description: Lemma for mthmb 35613. (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 2731	. . . . 5 ⊢ (mPPSt‘𝑇) = (mPPSt‘𝑇)
3		mthmb.u	. . . . 5 ⊢ 𝑈 = (mThm‘𝑇)
4	1, 2, 3	mthmval 35607	. . . 4 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ (mPPSt‘𝑇)))
5	4	eleq2i 2823	. . 3 ⊢ (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡𝑅 “ (𝑅 “ (mPPSt‘𝑇))))
6		eqid 2731	. . . . . 6 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
7	6, 1	msrf 35574	. . . . 5 ⊢ 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
8		ffn 6651	. . . . 5 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
9	7, 8	ax-mp 5	. . . 4 ⊢ 𝑅 Fn (mPreSt‘𝑇)
10		elpreima 6991	. . . 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 2819	. . . 4 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) ↔ (𝑅‘𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))))
14		ffun 6654	. . . . . . 7 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → Fun 𝑅)
15	7, 14	ax-mp 5	. . . . . 6 ⊢ Fun 𝑅
16		fvelima 6887	. . . . . 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 35609	. . . . . 6 ⊢ ((𝑥 ∈ (mPPSt‘𝑇) ∧ (𝑅‘𝑥) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈)
19	18	rexlimiva 3125	. . . . 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 2111  ∃wrex 3056  ^◡ccnv 5615   “ cima 5619  Fun wfun 6475   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  mPreStcmpst 35505  mStRedcmsr 35506  mPPStcmpps 35510  mThmcmthm 35511

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-ot 4585  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-1st 7921  df-2nd 7922  df-mpst 35525  df-msr 35526  df-mpps 35530  df-mthm 35531

This theorem is referenced by:  mthmb  35613