Theorem msrrcl 35579

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 35578	. . . 4 ⊢ 𝑅:𝑃⟶𝑃
4	3	ffvelcdmi 7011	. . 3 ⊢ (𝑋 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃)
5	4	a1i 11	. 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃))
6	3	ffvelcdmi 7011	. . 3 ⊢ (𝑌 ∈ 𝑃 → (𝑅‘𝑌) ∈ 𝑃)
7		eleq1 2819	. . 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 6657	. . . . . 6 ⊢ dom 𝑅 = 𝑃
10		0nelxp 5645	. . . . . . 7 ⊢ ¬ ∅ ∈ ((V × V) × V)
11	1	mpstssv 35575	. . . . . . . 8 ⊢ 𝑃 ⊆ ((V × V) × V)
12	11	sseli 3925	. . . . . . 7 ⊢ (∅ ∈ 𝑃 → ∅ ∈ ((V × V) × V))
13	10, 12	mto 197	. . . . . 6 ⊢ ¬ ∅ ∈ 𝑃
14	9, 13	ndmfvrcl 6850	. . . . 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 6850	. . . . 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 1541   ∈ wcel 2111  Vcvv 3436  ∅c0 4278   × cxp 5609  ‘cfv 6476  mPreStcmpst 35509  mStRedcmsr 35510

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-ot 4580  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-1st 7916  df-2nd 7917  df-mpst 35529  df-msr 35530

This theorem is referenced by:  elmthm  35612