Theorem mthmval 32822

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 6670	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3		mthmval.r	. . . . . . 7 ⊢ 𝑅 = (mStRed‘𝑇)
4	2, 3	syl6eqr 2874	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
5	4	cnveqd 5746	. . . . 5 ⊢ (𝑡 = 𝑇 → ◡(mStRed‘𝑡) = ◡𝑅)
6		fveq2 6670	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇))
7		mthmval.j	. . . . . . 7 ⊢ 𝐽 = (mPPSt‘𝑇)
8	6, 7	syl6eqr 2874	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽)
9	4, 8	imaeq12d 5930	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅 “ 𝐽))
10	5, 9	imaeq12d 5930	. . . 4 ⊢ (𝑡 = 𝑇 → (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (◡𝑅 “ (𝑅 “ 𝐽)))
11		df-mthm 32746	. . . 4 ⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
12		fvex 6683	. . . . . 6 ⊢ (mStRed‘𝑡) ∈ V
13	12	cnvex 7630	. . . . 5 ⊢ ◡(mStRed‘𝑡) ∈ V
14		imaexg 7620	. . . . 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 6773	. . 3 ⊢ (𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)))
17		0ima 5946	. . . . 5 ⊢ (∅ “ (𝑅 “ 𝐽)) = ∅
18	17	eqcomi 2830	. . . 4 ⊢ ∅ = (∅ “ (𝑅 “ 𝐽))
19		fvprc 6663	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = ∅)
20		fvprc 6663	. . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
21	3, 20	syl5eq 2868	. . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
22	21	cnveqd 5746	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ◡∅)
23		cnv0 5999	. . . . . 6 ⊢ ◡∅ = ∅
24	22, 23	syl6eq 2872	. . . . 5 ⊢ (¬ 𝑇 ∈ V → ◡𝑅 = ∅)
25	24	imaeq1d 5928	. . . 4 ⊢ (¬ 𝑇 ∈ V → (◡𝑅 “ (𝑅 “ 𝐽)) = (∅ “ (𝑅 “ 𝐽)))
26	18, 19, 25	3eqtr4a 2882	. . 3 ⊢ (¬ 𝑇 ∈ V → (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽)))
27	16, 26	pm2.61i 184	. 2 ⊢ (mThm‘𝑇) = (◡𝑅 “ (𝑅 “ 𝐽))
28	1, 27	eqtri 2844	1 ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽))

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ∅c0 4291  ^◡ccnv 5554   “ cima 5558  ‘cfv 6355  mStRedcmsr 32721  mPPStcmpps 32725  mThmcmthm 32726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fv 6363  df-mthm 32746

This theorem is referenced by:  elmthm  32823  mthmsta  32825  mthmblem  32827