Theorem elmthm 31811

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 31810	. . 3 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽))
5	4	eleq2i 2842	. 2 ⊢ (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡𝑅 “ (𝑅 “ 𝐽)))
6		eqid 2771	. . . . 5 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
7	6, 1	msrf 31777	. . . 4 ⊢ 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
8		ffn 6185	. . . 4 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
9	7, 8	ax-mp 5	. . 3 ⊢ 𝑅 Fn (mPreSt‘𝑇)
10		elpreima 6480	. . 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 31809	. . . . 5 ⊢ 𝐽 ⊆ (mPreSt‘𝑇)
13		fvelimab 6395	. . . . 5 ⊢ ((𝑅 Fn (mPreSt‘𝑇) ∧ 𝐽 ⊆ (mPreSt‘𝑇)) → ((𝑅‘𝑋) ∈ (𝑅 “ 𝐽) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)))
14	9, 12, 13	mp2an 672	. . . 4 ⊢ ((𝑅‘𝑋) ∈ (𝑅 “ 𝐽) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))
15	14	anbi2i 609	. . 3 ⊢ ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)))
16	12	sseli 3748	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ∈ (mPreSt‘𝑇))
17	6, 1	msrrcl 31778	. . . . . 6 ⊢ ((𝑅‘𝑥) = (𝑅‘𝑋) → (𝑥 ∈ (mPreSt‘𝑇) ↔ 𝑋 ∈ (mPreSt‘𝑇)))
18	16, 17	syl5ibcom 235	. . . . 5 ⊢ (𝑥 ∈ 𝐽 → ((𝑅‘𝑥) = (𝑅‘𝑋) → 𝑋 ∈ (mPreSt‘𝑇)))
19	18	rexlimiv 3175	. . . 4 ⊢ (∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋) → 𝑋 ∈ (mPreSt‘𝑇))
20	19	pm4.71ri 550	. . 3 ⊢ (∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)))
21	15, 20	bitr4i 267	. 2 ⊢ ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅‘𝑋) ∈ (𝑅 “ 𝐽)) ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))
22	5, 11, 21	3bitri 286	1 ⊢ (𝑋 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))

Syntax hints:   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∃wrex 3062   ⊆ wss 3723  ^◡ccnv 5248   “ cima 5252   Fn wfn 6026  ⟶wf 6027  ‘cfv 6031  mPreStcmpst 31708  mStRedcmsr 31709  mPPStcmpps 31713  mThmcmthm 31714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-ot 4325  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-1st 7315  df-2nd 7316  df-mpst 31728  df-msr 31729  df-mpps 31733  df-mthm 31734

This theorem is referenced by:  mthmi  31812  mthmpps  31817