Theorem elmthm 33538

Description: A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. (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
elmthm	⊢ (𝑋 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑅   𝑥,𝑇   𝑥,𝑋

Allowed substitution hint:   𝑈(𝑥)

Proof of Theorem elmthm

Step	Hyp	Ref	Expression
1		mthmval.r	. . . 4 ⊢ 𝑅 = (mStRed‘𝑇)
2		mthmval.j	. . . 4 ⊢ 𝐽 = (mPPSt‘𝑇)
3		mthmval.u	. . . 4 ⊢ 𝑈 = (mThm‘𝑇)
4	1, 2, 3	mthmval 33537	. . 3 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽))
5	4	eleq2i 2830	. 2 ⊢ (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡𝑅 “ (𝑅 “ 𝐽)))
6		eqid 2738	. . . . 5 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
7	6, 1	msrf 33504	. . . 4 ⊢ 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
8		ffn 6600	. . . 4 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
9	7, 8	ax-mp 5	. . 3 ⊢ 𝑅 Fn (mPreSt‘𝑇)
10		elpreima 6935	. . 3 ⊢ (𝑅 Fn (mPreSt‘𝑇) → (𝑋 ∈ (◡𝑅 “ (𝑅 “ 𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽))))
11	9, 10	ax-mp 5	. 2 ⊢ (𝑋 ∈ (◡𝑅 “ (𝑅 “ 𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽)))
12	6, 2	mppspst 33536	. . . . 5 ⊢ 𝐽 ⊆ (mPreSt‘𝑇)
13		fvelimab 6841	. . . . 5 ⊢ ((𝑅 Fn (mPreSt‘𝑇) ∧ 𝐽 ⊆ (mPreSt‘𝑇)) → ((𝑅‘𝑋) ∈ (𝑅 “ 𝐽) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)))
14	9, 12, 13	mp2an 689	. . . 4 ⊢ ((𝑅‘𝑋) ∈ (𝑅 “ 𝐽) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))
15	14	anbi2i 623	. . 3 ⊢ ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)))
16	12	sseli 3917	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ∈ (mPreSt‘𝑇))
17	6, 1	msrrcl 33505	. . . . . 6 ⊢ ((𝑅‘𝑥) = (𝑅‘𝑋) → (𝑥 ∈ (mPreSt‘𝑇) ↔ 𝑋 ∈ (mPreSt‘𝑇)))
18	16, 17	syl5ibcom 244	. . . . 5 ⊢ (𝑥 ∈ 𝐽 → ((𝑅‘𝑥) = (𝑅‘𝑋) → 𝑋 ∈ (mPreSt‘𝑇)))
19	18	rexlimiv 3209	. . . 4 ⊢ (∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋) → 𝑋 ∈ (mPreSt‘𝑇))
20	19	pm4.71ri 561	. . 3 ⊢ (∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)))
21	15, 20	bitr4i 277	. 2 ⊢ ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽)) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))
22	5, 11, 21	3bitri 297	1 ⊢ (𝑋 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ⊆ wss 3887  ^◡ccnv 5588   “ cima 5592   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  mPreStcmpst 33435  mStRedcmsr 33436  mPPStcmpps 33440  mThmcmthm 33441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-ot 4570  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-1st 7831  df-2nd 7832  df-mpst 33455  df-msr 33456  df-mpps 33460  df-mthm 33461

This theorem is referenced by:  mthmi  33539  mthmpps  33544