Theorem mthmval 31801

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 6354	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3		mthmval.r	. . . . . . 7 ⊢ 𝑅 = (mStRed‘𝑇)
4	2, 3	syl6eqr 2813	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
5	4	cnveqd 5454	. . . . 5 ⊢ (𝑡 = 𝑇 → ◡(mStRed‘𝑡) = ◡𝑅)
6		fveq2 6354	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇))
7		mthmval.j	. . . . . . 7 ⊢ 𝐽 = (mPPSt‘𝑇)
8	6, 7	syl6eqr 2813	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽)
9	4, 8	imaeq12d 5626	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅 “ 𝐽))
10	5, 9	imaeq12d 5626	. . . 4 ⊢ (𝑡 = 𝑇 → (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (◡𝑅 “ (𝑅 “ 𝐽)))
11		df-mthm 31725	. . . 4 ⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
12		fvex 6364	. . . . . 6 ⊢ (mStRed‘𝑡) ∈ V
13	12	cnvex 7280	. . . . 5 ⊢ ◡(mStRed‘𝑡) ∈ V
14		imaexg 7270	. . . . 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 6451	. . 3 ⊢ (𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)))
17		0ima 5641	. . . . 5 ⊢ (∅ “ (𝑅 “ 𝐽)) = ∅
18	17	eqcomi 2770	. . . 4 ⊢ ∅ = (∅ “ (𝑅 “ 𝐽))
19		fvprc 6348	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = ∅)
20		fvprc 6348	. . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
21	3, 20	syl5eq 2807	. . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
22	21	cnveqd 5454	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ◡∅)
23		cnv0 5694	. . . . . 6 ⊢ ◡∅ = ∅
24	22, 23	syl6eq 2811	. . . . 5 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ∅)
25	24	imaeq1d 5624	. . . 4 ⊢ (¬ 𝑇 ∈ V → (◡𝑅 “ (𝑅 “ 𝐽)) = (∅ “ (𝑅 “ 𝐽)))
26	18, 19, 25	3eqtr4a 2821	. . 3 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)))
27	16, 26	pm2.61i 176	. 2 ⊢ (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽))
28	1, 27	eqtri 2783	1 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽))

Syntax hints:  ¬ wn 3   = wceq 1632   ∈ wcel 2140  Vcvv 3341  ∅c0 4059  ^◡ccnv 5266   “ cima 5270  ‘cfv 6050  mStRedcmsr 31700  mPPStcmpps 31704  mThmcmthm 31705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fv 6058  df-mthm 31725

This theorem is referenced by:  elmthm  31802  mthmsta  31804  mthmblem  31806