Theorem elmthm 36427

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 36426	. . 3 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽))
5	4	eleq2i 2852	. 2 ⊢ (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡𝑅 “ (𝑅 “ 𝐽)))
6		eqid 2760	. . . . 5 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
7	6, 1	msrf 36393	. . . 4 ⊢ 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
8		ffn 6709	. . . 4 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
9	7, 8	ax-mp 5	. . 3 ⊢ 𝑅 Fn (mPreSt‘𝑇)
10		elpreima 7057	. . 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 36425	. . . . 5 ⊢ 𝐽 ⊆ (mPreSt‘𝑇)
13		fvelimab 6957	. . . . 5 ⊢ ((𝑅 Fn (mPreSt‘𝑇) ∧ 𝐽 ⊆ (mPreSt‘𝑇)) → ((𝑅‘𝑋) ∈ (𝑅 “ 𝐽) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)))
14	9, 12, 13	mp2an 705	. . . 4 ⊢ ((𝑅‘𝑋) ∈ (𝑅 “ 𝐽) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))
15	14	anbi2i 635	. . 3 ⊢ ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)))
16	12	sseli 3904	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ∈ (mPreSt‘𝑇))
17	6, 1	msrrcl 36394	. . . . . 6 ⊢ ((𝑅‘𝑥) = (𝑅‘𝑋) → (𝑥 ∈ (mPreSt‘𝑇) ↔ 𝑋 ∈ (mPreSt‘𝑇)))
18	16, 17	syl5ibcom 248	. . . . 5 ⊢ (𝑥 ∈ 𝐽 → ((𝑅‘𝑥) = (𝑅‘𝑋) → 𝑋 ∈ (mPreSt‘𝑇)))
19	18	rexlimiv 3156	. . . 4 ⊢ (∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋) → 𝑋 ∈ (mPreSt‘𝑇))
20	19	pm4.71ri 570	. . 3 ⊢ (∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)))
21	15, 20	bitr4i 281	. 2 ⊢ ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽)) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))
22	5, 11, 21	3bitri 300	1 ⊢ (𝑋 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895  ^◡ccnv 5649   “ cima 5653   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  mPreStcmpst 36324  mStRedcmsr 36325  mPPStcmpps 36329  mThmcmthm 36330

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-ot 4858  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-1st 8001  df-2nd 8002  df-mpst 36344  df-msr 36345  df-mpps 36349  df-mthm 36350

This theorem is used by:  mthmi  36428  mthmpps  36433