Theorem msrrcl 35511

Description: If 𝑋 and 𝑌 have the same reduct, then one is a pre-statement iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mpstssv.p	⊢ 𝑃 = (mPreSt‘𝑇)
msrf.r	⊢ 𝑅 = (mStRed‘𝑇)

Assertion

Ref	Expression
msrrcl	⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃))

Proof of Theorem msrrcl

Step	Hyp	Ref	Expression
1		mpstssv.p	. . . . 5 ⊢ 𝑃 = (mPreSt‘𝑇)
2		msrf.r	. . . . 5 ⊢ 𝑅 = (mStRed‘𝑇)
3	1, 2	msrf 35510	. . . 4 ⊢ 𝑅:𝑃⟶𝑃
4	3	ffvelcdmi 7117	. . 3 ⊢ (𝑋 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃)
5	4	a1i 11	. 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃))
6	3	ffvelcdmi 7117	. . 3 ⊢ (𝑌 ∈ 𝑃 → (𝑅‘𝑌) ∈ 𝑃)
7		eleq1 2832	. . 3 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ 𝑃 ↔ (𝑅‘𝑌) ∈ 𝑃))
8	6, 7	imbitrrid 246	. 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑌 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃))
9	3	fdmi 6758	. . . . . 6 ⊢ dom 𝑅 = 𝑃
10		0nelxp 5734	. . . . . . 7 ⊢ ¬ ∅ ∈ ((V × V) × V)
11	1	mpstssv 35507	. . . . . . . 8 ⊢ 𝑃 ⊆ ((V × V) × V)
12	11	sseli 4004	. . . . . . 7 ⊢ (∅ ∈ 𝑃 → ∅ ∈ ((V × V) × V))
13	10, 12	mto 197	. . . . . 6 ⊢ ¬ ∅ ∈ 𝑃
14	9, 13	ndmfvrcl 6956	. . . . 5 ⊢ ((𝑅‘𝑋) ∈ 𝑃 → 𝑋 ∈ 𝑃)
15	14	adantl 481	. . . 4 ⊢ (((𝑅‘𝑋) = (𝑅‘𝑌) ∧ (𝑅‘𝑋) ∈ 𝑃) → 𝑋 ∈ 𝑃)
16	7	biimpa 476	. . . . 5 ⊢ (((𝑅‘𝑋) = (𝑅‘𝑌) ∧ (𝑅‘𝑋) ∈ 𝑃) → (𝑅‘𝑌) ∈ 𝑃)
17	9, 13	ndmfvrcl 6956	. . . . 5 ⊢ ((𝑅‘𝑌) ∈ 𝑃 → 𝑌 ∈ 𝑃)
18	16, 17	syl 17	. . . 4 ⊢ (((𝑅‘𝑋) = (𝑅‘𝑌) ∧ (𝑅‘𝑋) ∈ 𝑃) → 𝑌 ∈ 𝑃)
19	15, 18	2thd 265	. . 3 ⊢ (((𝑅‘𝑋) = (𝑅‘𝑌) ∧ (𝑅‘𝑋) ∈ 𝑃) → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃))
20	19	ex 412	. 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ 𝑃 → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃)))
21	5, 8, 20	pm5.21ndd 379	1 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352   × cxp 5698  ‘cfv 6573  mPreStcmpst 35441  mStRedcmsr 35442

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-rep 5303  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-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  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-ot 4657  df-uni 4932  df-iun 5017  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-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-1st 8030  df-2nd 8031  df-mpst 35461  df-msr 35462

This theorem is referenced by:  elmthm  35544