Theorem mthmval 34256

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 6847	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3		mthmval.r	. . . . . . 7 ⊢ 𝑅 = (mStRed‘𝑇)
4	2, 3	eqtr4di 2789	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
5	4	cnveqd 5836	. . . . 5 ⊢ (𝑡 = 𝑇 → ◡(mStRed‘𝑡) = ◡𝑅)
6		fveq2 6847	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇))
7		mthmval.j	. . . . . . 7 ⊢ 𝐽 = (mPPSt‘𝑇)
8	6, 7	eqtr4di 2789	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽)
9	4, 8	imaeq12d 6019	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅 “ 𝐽))
10	5, 9	imaeq12d 6019	. . . 4 ⊢ (𝑡 = 𝑇 → (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (◡𝑅 “ (𝑅 “ 𝐽)))
11		df-mthm 34180	. . . 4 ⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
12		fvex 6860	. . . . . 6 ⊢ (mStRed‘𝑡) ∈ V
13	12	cnvex 7867	. . . . 5 ⊢ ◡(mStRed‘𝑡) ∈ V
14		imaexg 7857	. . . . 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 6958	. . 3 ⊢ (𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)))
17		0ima 6035	. . . . 5 ⊢ (∅ “ (𝑅 “ 𝐽)) = ∅
18	17	eqcomi 2740	. . . 4 ⊢ ∅ = (∅ “ (𝑅 “ 𝐽))
19		fvprc 6839	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = ∅)
20		fvprc 6839	. . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
21	3, 20	eqtrid 2783	. . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
22	21	cnveqd 5836	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ◡∅)
23		cnv0 6098	. . . . . 6 ⊢ ◡∅ = ∅
24	22, 23	eqtrdi 2787	. . . . 5 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ∅)
25	24	imaeq1d 6017	. . . 4 ⊢ (¬ 𝑇 ∈ V → (◡𝑅 “ (𝑅 “ 𝐽)) = (∅ “ (𝑅 “ 𝐽)))
26	18, 19, 25	3eqtr4a 2797	. . 3 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)))
27	16, 26	pm2.61i 182	. 2 ⊢ (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽))
28	1, 27	eqtri 2759	1 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽))

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2106  Vcvv 3446  ∅c0 4287  ^◡ccnv 5637   “ cima 5641  ‘cfv 6501  mStRedcmsr 34155  mPPStcmpps 34159  mThmcmthm 34160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fv 6509  df-mthm 34180

This theorem is referenced by:  elmthm  34257  mthmsta  34259  mthmblem  34261