Theorem msrrcl 35532

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 35531	. . . 4 ⊢ 𝑅:𝑃⟶𝑃
4	3	ffvelcdmi 7062	. . 3 ⊢ (𝑋 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃)
5	4	a1i 11	. 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃))
6	3	ffvelcdmi 7062	. . 3 ⊢ (𝑌 ∈ 𝑃 → (𝑅‘𝑌) ∈ 𝑃)
7		eleq1 2817	. . 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 6706	. . . . . 6 ⊢ dom 𝑅 = 𝑃
10		0nelxp 5680	. . . . . . 7 ⊢ ¬ ∅ ∈ ((V × V) × V)
11	1	mpstssv 35528	. . . . . . . 8 ⊢ 𝑃 ⊆ ((V × V) × V)
12	11	sseli 3950	. . . . . . 7 ⊢ (∅ ∈ 𝑃 → ∅ ∈ ((V × V) × V))
13	10, 12	mto 197	. . . . . 6 ⊢ ¬ ∅ ∈ 𝑃
14	9, 13	ndmfvrcl 6901	. . . . 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 6901	. . . . 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 1540   ∈ wcel 2109  Vcvv 3455  ∅c0 4304   × cxp 5644  ‘cfv 6519  mPreStcmpst 35462  mStRedcmsr 35463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-ot 4606  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-1st 7977  df-2nd 7978  df-mpst 35482  df-msr 35483

This theorem is referenced by:  elmthm  35565