Theorem mthmval 35543

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 6920	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3		mthmval.r	. . . . . . 7 ⊢ 𝑅 = (mStRed‘𝑇)
4	2, 3	eqtr4di 2798	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
5	4	cnveqd 5900	. . . . 5 ⊢ (𝑡 = 𝑇 → ◡(mStRed‘𝑡) = ◡𝑅)
6		fveq2 6920	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇))
7		mthmval.j	. . . . . . 7 ⊢ 𝐽 = (mPPSt‘𝑇)
8	6, 7	eqtr4di 2798	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽)
9	4, 8	imaeq12d 6090	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅 “ 𝐽))
10	5, 9	imaeq12d 6090	. . . 4 ⊢ (𝑡 = 𝑇 → (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (◡𝑅 “ (𝑅 “ 𝐽)))
11		df-mthm 35467	. . . 4 ⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
12		fvex 6933	. . . . . 6 ⊢ (mStRed‘𝑡) ∈ V
13	12	cnvex 7965	. . . . 5 ⊢ ◡(mStRed‘𝑡) ∈ V
14		imaexg 7953	. . . . 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 7034	. . 3 ⊢ (𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)))
17		0ima 6107	. . . . 5 ⊢ (∅ “ (𝑅 “ 𝐽)) = ∅
18	17	eqcomi 2749	. . . 4 ⊢ ∅ = (∅ “ (𝑅 “ 𝐽))
19		fvprc 6912	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = ∅)
20		fvprc 6912	. . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
21	3, 20	eqtrid 2792	. . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
22	21	cnveqd 5900	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ◡∅)
23		cnv0 6172	. . . . . 6 ⊢ ◡∅ = ∅
24	22, 23	eqtrdi 2796	. . . . 5 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ∅)
25	24	imaeq1d 6088	. . . 4 ⊢ (¬ 𝑇 ∈ V → (◡𝑅 “ (𝑅 “ 𝐽)) = (∅ “ (𝑅 “ 𝐽)))
26	18, 19, 25	3eqtr4a 2806	. . 3 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)))
27	16, 26	pm2.61i 182	. 2 ⊢ (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽))
28	1, 27	eqtri 2768	1 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽))

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  ^◡ccnv 5699   “ cima 5703  ‘cfv 6573  mStRedcmsr 35442  mPPStcmpps 35446  mThmcmthm 35447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-mthm 35467

This theorem is referenced by:  elmthm  35544  mthmsta  35546  mthmblem  35548