mthmpps - Mathbox for Mario Carneiro

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Theorem mthmpps 30733

Description: Given a theorem, there is an explicitly definable witnessing provable pre-statement for the provability of the theorem. (However, this pre-statement requires infinitely many dv conditions, which is sometimes inconvenient.) (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypotheses**
Ref	Expression
mthmpps.r	⊢ 𝑅 = (mStRed‘𝑇)
mthmpps.j	⊢ 𝐽 = (mPPSt‘𝑇)
mthmpps.u	⊢ 𝑈 = (mThm‘𝑇)
mthmpps.d	⊢ 𝐷 = (mDV‘𝑇)
mthmpps.v	⊢ 𝑉 = (mVars‘𝑇)
mthmpps.z	⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))
mthmpps.m	⊢ 𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))

**Assertion**
Ref	Expression
mthmpps	⊢ (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))

Proof of Theorem mthmpps Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mthmpps.m	. . . . . . . 8 ⊢ 𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))
2		mthmpps.u	. . . . . . . . . . . . . 14 ⊢ 𝑈 = (mThm‘𝑇)
3		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
4	2, 3	mthmsta 30729	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ (mPreSt‘𝑇)
5		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
6	4, 5	sseldi 3566	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
7		mthmpps.d	. . . . . . . . . . . . 13 ⊢ 𝐷 = (mDV‘𝑇)
8		eqid 2610	. . . . . . . . . . . . 13 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
9	7, 8, 3	elmpst 30687	. . . . . . . . . . . 12 ⊢ (⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝐶 ⊆ 𝐷 ∧ ^◡𝐶 = 𝐶) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
10	6, 9	sylib 207	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ((𝐶 ⊆ 𝐷 ∧ ^◡𝐶 = 𝐶) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
11	10	simp1d 1066	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶 ⊆ 𝐷 ∧ ^◡𝐶 = 𝐶))
12	11	simpld 474	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶 ⊆ 𝐷)
13		difssd 3700	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) ⊆ 𝐷)
14	12, 13	unssd 3751	. . . . . . . 8 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ⊆ 𝐷)
15	1, 14	syl5eqss 3612	. . . . . . 7 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀 ⊆ 𝐷)
16	11	simprd 478	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ^◡𝐶 = 𝐶)
17		cnvdif 5458	. . . . . . . . . . 11 ⊢ ^◡(𝐷 ∖ (𝑍 × 𝑍)) = (^◡𝐷 ∖ ^◡(𝑍 × 𝑍))
18		cnvdif 5458	. . . . . . . . . . . . . 14 ⊢ ^◡(((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (^◡((mVR‘𝑇) × (mVR‘𝑇)) ∖ ^◡ I )
19		cnvxp 5470	. . . . . . . . . . . . . . 15 ⊢ ^◡((mVR‘𝑇) × (mVR‘𝑇)) = ((mVR‘𝑇) × (mVR‘𝑇))
20		cnvi 5456	. . . . . . . . . . . . . . 15 ⊢ ^◡ I = I
21	19, 20	difeq12i 3688	. . . . . . . . . . . . . 14 ⊢ (^◡((mVR‘𝑇) × (mVR‘𝑇)) ∖ ^◡ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
22	18, 21	eqtri 2632	. . . . . . . . . . . . 13 ⊢ ^◡(((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
23		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
24	23, 7	mdvval 30655	. . . . . . . . . . . . . 14 ⊢ 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
25	24	cnveqi 5219	. . . . . . . . . . . . 13 ⊢ ^◡𝐷 = ^◡(((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
26	22, 25, 24	3eqtr4i 2642	. . . . . . . . . . . 12 ⊢ ^◡𝐷 = 𝐷
27		cnvxp 5470	. . . . . . . . . . . 12 ⊢ ^◡(𝑍 × 𝑍) = (𝑍 × 𝑍)
28	26, 27	difeq12i 3688	. . . . . . . . . . 11 ⊢ (^◡𝐷 ∖ ^◡(𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
29	17, 28	eqtri 2632	. . . . . . . . . 10 ⊢ ^◡(𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
30	29	a1i 11	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ^◡(𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍)))
31	16, 30	uneq12d 3730	. . . . . . . 8 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (^◡𝐶 ∪ ^◡(𝐷 ∖ (𝑍 × 𝑍))) = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
32	1	cnveqi 5219	. . . . . . . . 9 ⊢ ^◡𝑀 = ^◡(𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))
33		cnvun 5457	. . . . . . . . 9 ⊢ ^◡(𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) = (^◡𝐶 ∪ ^◡(𝐷 ∖ (𝑍 × 𝑍)))
34	32, 33	eqtri 2632	. . . . . . . 8 ⊢ ^◡𝑀 = (^◡𝐶 ∪ ^◡(𝐷 ∖ (𝑍 × 𝑍)))
35	31, 34, 1	3eqtr4g 2669	. . . . . . 7 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ^◡𝑀 = 𝑀)
36	15, 35	jca 553	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀 ⊆ 𝐷 ∧ ^◡𝑀 = 𝑀))
37	10	simp2d 1067	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
38	10	simp3d 1068	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (mEx‘𝑇))
39	7, 8, 3	elmpst 30687	. . . . . 6 ⊢ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀 ⊆ 𝐷 ∧ ^◡𝑀 = 𝑀) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
40	36, 37, 38, 39	syl3anbrc 1239	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
41		mthmpps.r	. . . . . . . 8 ⊢ 𝑅 = (mStRed‘𝑇)
42		mthmpps.j	. . . . . . . 8 ⊢ 𝐽 = (mPPSt‘𝑇)
43	41, 42, 2	elmthm 30727	. . . . . . 7 ⊢ (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
44	5, 43	sylib 207	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
45		eqid 2610	. . . . . . . 8 ⊢ (mCls‘𝑇) = (mCls‘𝑇)
46		simpll 786	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑇 ∈ mFS)
47	15	adantr 480	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑀 ⊆ 𝐷)
48	37	simpld 474	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐻 ⊆ (mEx‘𝑇))
49	48	adantr 480	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻 ⊆ (mEx‘𝑇))
50	3, 42	mppspst 30725	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 ⊆ (mPreSt‘𝑇)
51		simprl 790	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 ∈ 𝐽)
52	50, 51	sseldi 3566	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 ∈ (mPreSt‘𝑇))
53	3	mpst123 30691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (mPreSt‘𝑇) → 𝑥 = ⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩)
54	52, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 = ⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩)
55	54	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘𝑥) = (𝑅‘⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩))
56		simprr 792	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
57	55, 56	eqtr3d 2646	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
58	54, 52	eqeltrrd 2689	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩ ∈ (mPreSt‘𝑇))
59		mthmpps.v	. . . . . . . . . . . . . . . . 17 ⊢ 𝑉 = (mVars‘𝑇)
60		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) = ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)}))
61	59, 3, 41, 60	msrval 30689	. . . . . . . . . . . . . . . 16 ⊢ (⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩) = ⟨((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩)
62	58, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩) = ⟨((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩)
63		mthmpps.z	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))
64	59, 3, 41, 63	msrval 30689	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨𝐶, 𝐻, 𝐴⟩) = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
65	6, 64	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝐶, 𝐻, 𝐴⟩) = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
66	65	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘⟨𝐶, 𝐻, 𝐴⟩) = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
67	57, 62, 66	3eqtr3d 2652	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
68		fvex 6113	. . . . . . . . . . . . . . . 16 ⊢ (1^st ‘(1^st ‘𝑥)) ∈ V
69	68	inex1 4727	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))) ∈ V
70		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (2^nd ‘(1^st ‘𝑥)) ∈ V
71		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (2^nd ‘𝑥) ∈ V
72	69, 70, 71	otth 4879	. . . . . . . . . . . . . 14 ⊢ (⟨((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ↔ (((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))) = (𝐶 ∩ (𝑍 × 𝑍)) ∧ (2^nd ‘(1^st ‘𝑥)) = 𝐻 ∧ (2^nd ‘𝑥) = 𝐴))
73	67, 72	sylib 207	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))) = (𝐶 ∩ (𝑍 × 𝑍)) ∧ (2^nd ‘(1^st ‘𝑥)) = 𝐻 ∧ (2^nd ‘𝑥) = 𝐴))
74	73	simp1d 1066	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))) = (𝐶 ∩ (𝑍 × 𝑍)))
75	73	simp2d 1067	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2^nd ‘(1^st ‘𝑥)) = 𝐻)
76	73	simp3d 1068	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2^nd ‘𝑥) = 𝐴)
77	76	sneqd 4137	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → {(2^nd ‘𝑥)} = {𝐴})
78	75, 77	uneq12d 3730	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)}) = (𝐻 ∪ {𝐴}))
79	78	imaeq2d 5385	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) = (𝑉 “ (𝐻 ∪ {𝐴})))
80	79	unieqd 4382	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) = ∪ (𝑉 “ (𝐻 ∪ {𝐴})))
81	80, 63	syl6eqr 2662	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) = 𝑍)
82	81	sqxpeqd 5065	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)}))) = (𝑍 × 𝑍))
83	82	ineq2d 3776	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))) = ((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)))
84	74, 83	eqtr3d 2646	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝐶 ∩ (𝑍 × 𝑍)) = ((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)))
85		inss1 3795	. . . . . . . . . . 11 ⊢ (𝐶 ∩ (𝑍 × 𝑍)) ⊆ 𝐶
86	84, 85	syl6eqssr 3619	. . . . . . . . . 10 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)) ⊆ 𝐶)
87		eqidd 2611	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑥)))
88	87, 75, 76	oteq123d 4355	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩ = ⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩)
89	54, 88	eqtrd 2644	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 = ⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩)
90	89, 52	eqeltrrd 2689	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
91	7, 8, 3	elmpst 30687	. . . . . . . . . . . . . 14 ⊢ (⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ (((1^st ‘(1^st ‘𝑥)) ⊆ 𝐷 ∧ ^◡(1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑥))) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
92	91	simp1bi 1069	. . . . . . . . . . . . 13 ⊢ (⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → ((1^st ‘(1^st ‘𝑥)) ⊆ 𝐷 ∧ ^◡(1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑥))))
93	92	simpld 474	. . . . . . . . . . . 12 ⊢ (⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → (1^st ‘(1^st ‘𝑥)) ⊆ 𝐷)
94	90, 93	syl 17	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1^st ‘(1^st ‘𝑥)) ⊆ 𝐷)
95	94	ssdifd 3708	. . . . . . . . . 10 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1^st ‘(1^st ‘𝑥)) ∖ (𝑍 × 𝑍)) ⊆ (𝐷 ∖ (𝑍 × 𝑍)))
96		unss12 3747	. . . . . . . . . 10 ⊢ ((((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)) ⊆ 𝐶 ∧ ((1^st ‘(1^st ‘𝑥)) ∖ (𝑍 × 𝑍)) ⊆ (𝐷 ∖ (𝑍 × 𝑍))) → (((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1^st ‘(1^st ‘𝑥)) ∖ (𝑍 × 𝑍))) ⊆ (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
97	86, 95, 96	syl2anc 691	. . . . . . . . 9 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1^st ‘(1^st ‘𝑥)) ∖ (𝑍 × 𝑍))) ⊆ (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
98		inundif 3998	. . . . . . . . . 10 ⊢ (((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1^st ‘(1^st ‘𝑥)) ∖ (𝑍 × 𝑍))) = (1^st ‘(1^st ‘𝑥))
99	98	eqcomi 2619	. . . . . . . . 9 ⊢ (1^st ‘(1^st ‘𝑥)) = (((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1^st ‘(1^st ‘𝑥)) ∖ (𝑍 × 𝑍)))
100	97, 99, 1	3sstr4g 3609	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1^st ‘(1^st ‘𝑥)) ⊆ 𝑀)
101		ssid 3587	. . . . . . . . 9 ⊢ 𝐻 ⊆ 𝐻
102	101	a1i 11	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻 ⊆ 𝐻)
103	7, 8, 45, 46, 47, 49, 100, 102	ss2mcls 30719	. . . . . . 7 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1^st ‘(1^st ‘𝑥))(mCls‘𝑇)𝐻) ⊆ (𝑀(mCls‘𝑇)𝐻))
104	89, 51	eqeltrrd 2689	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽)
105	3, 42, 45	elmpps 30724	. . . . . . . . 9 ⊢ (⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ ((1^st ‘(1^st ‘𝑥))(mCls‘𝑇)𝐻)))
106	105	simprbi 479	. . . . . . . 8 ⊢ (⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽 → 𝐴 ∈ ((1^st ‘(1^st ‘𝑥))(mCls‘𝑇)𝐻))
107	104, 106	syl 17	. . . . . . 7 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ ((1^st ‘(1^st ‘𝑥))(mCls‘𝑇)𝐻))
108	103, 107	sseldd 3569	. . . . . 6 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
109	44, 108	rexlimddv 3017	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
110	3, 42, 45	elmpps 30724	. . . . 5 ⊢ (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻)))
111	40, 109, 110	sylanbrc 695	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽)
112	1	ineq1i 3772	. . . . . . . 8 ⊢ (𝑀 ∩ (𝑍 × 𝑍)) = ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍))
113		indir 3834	. . . . . . . 8 ⊢ ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍)) = ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)))
114		incom 3767	. . . . . . . . . . 11 ⊢ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) = ((𝑍 × 𝑍) ∩ (𝐷 ∖ (𝑍 × 𝑍)))
115		disjdif 3992	. . . . . . . . . . 11 ⊢ ((𝑍 × 𝑍) ∩ (𝐷 ∖ (𝑍 × 𝑍))) = ∅
116	114, 115	eqtri 2632	. . . . . . . . . 10 ⊢ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) = ∅
117		0ss 3924	. . . . . . . . . 10 ⊢ ∅ ⊆ (𝐶 ∩ (𝑍 × 𝑍))
118	116, 117	eqsstri 3598	. . . . . . . . 9 ⊢ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍))
119		ssequn2 3748	. . . . . . . . 9 ⊢ (((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍)) ↔ ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍)))
120	118, 119	mpbi 219	. . . . . . . 8 ⊢ ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍))
121	112, 113, 120	3eqtri 2636	. . . . . . 7 ⊢ (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍))
122	121	a1i 11	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍)))
123	122	oteq1d 4352	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
124	59, 3, 41, 63	msrval 30689	. . . . . 6 ⊢ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
125	40, 124	syl 17	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
126	123, 125, 65	3eqtr4d 2654	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
127	111, 126	jca 553	. . 3 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)))
128	127	ex 449	. 2 ⊢ (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))
129	41, 42, 2	mthmi 30728	. 2 ⊢ ((⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
130	128, 129	impbid1 214	1 ⊢ (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 ⟨cotp 4133 ∪ cuni 4372 I cid 4948 × cxp 5036 ^◡ccnv 5037 “ cima 5041 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 Fincfn 7841 mVRcmvar 30612 mExcmex 30618 mDVcmdv 30619 mVarscmvrs 30620 mPreStcmpst 30624 mStRedcmsr 30625 mFScmfs 30627 mClscmcls 30628 mPPStcmpps 30629 mThmcmthm 30630

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-frmd 17209 df-mrex 30637 df-mex 30638 df-mdv 30639 df-mrsub 30641 df-msub 30642 df-mvh 30643 df-mpst 30644 df-msr 30645 df-msta 30646 df-mfs 30647 df-mcls 30648 df-mpps 30649 df-mthm 30650

This theorem is referenced by: (None)

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