Theorem mthmval 35560

Description: A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. Unlike the difference between pre-statement and statement, this application of the reduct is not necessarily trivial: there are theorems that are not themselves provable but are provable once enough "dummy variables" are introduced. (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
mthmval	⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽))

Proof of Theorem mthmval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mthmval.u	. 2 ⊢ 𝑈 = (mThm‘𝑇)
2		fveq2 6907	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3		mthmval.r	. . . . . . 7 ⊢ 𝑅 = (mStRed‘𝑇)
4	2, 3	eqtr4di 2793	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
5	4	cnveqd 5889	. . . . 5 ⊢ (𝑡 = 𝑇 → ◡(mStRed‘𝑡) = ◡𝑅)
6		fveq2 6907	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇))
7		mthmval.j	. . . . . . 7 ⊢ 𝐽 = (mPPSt‘𝑇)
8	6, 7	eqtr4di 2793	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽)
9	4, 8	imaeq12d 6081	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅 “ 𝐽))
10	5, 9	imaeq12d 6081	. . . 4 ⊢ (𝑡 = 𝑇 → (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (◡𝑅 “ (𝑅 “ 𝐽)))
11		df-mthm 35484	. . . 4 ⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
12		fvex 6920	. . . . . 6 ⊢ (mStRed‘𝑡) ∈ V
13	12	cnvex 7948	. . . . 5 ⊢ ◡(mStRed‘𝑡) ∈ V
14		imaexg 7936	. . . . 5 ⊢ (◡(mStRed‘𝑡) ∈ V → (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) ∈ V)
15	13, 14	ax-mp 5	. . . 4 ⊢ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) ∈ V
16	10, 11, 15	fvmpt3i 7021	. . 3 ⊢ (𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)))
17		0ima 6098	. . . . 5 ⊢ (∅ “ (𝑅 “ 𝐽)) = ∅
18	17	eqcomi 2744	. . . 4 ⊢ ∅ = (∅ “ (𝑅 “ 𝐽))
19		fvprc 6899	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = ∅)
20		fvprc 6899	. . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
21	3, 20	eqtrid 2787	. . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
22	21	cnveqd 5889	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ◡∅)
23		cnv0 6163	. . . . . 6 ⊢ ◡∅ = ∅
24	22, 23	eqtrdi 2791	. . . . 5 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ∅)
25	24	imaeq1d 6079	. . . 4 ⊢ (¬ 𝑇 ∈ V → (◡𝑅 “ (𝑅 “ 𝐽)) = (∅ “ (𝑅 “ 𝐽)))
26	18, 19, 25	3eqtr4a 2801	. . 3 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)))
27	16, 26	pm2.61i 182	. 2 ⊢ (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽))
28	1, 27	eqtri 2763	1 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽))

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∅c0 4339  ^◡ccnv 5688   “ cima 5692  ‘cfv 6563  mStRedcmsr 35459  mPPStcmpps 35463  mThmcmthm 35464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-mthm 35484

This theorem is referenced by:  elmthm  35561  mthmsta  35563  mthmblem  35565