Theorem mclspps 36436

Description: The closure is closed under application of provable pre-statements. (Compare mclsax 36421.) This theorem is what justifies the treatment of theorems as "equivalent" to axioms once they have been proven: the composition of one theorem in the proof of another yields a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mclspps.d	⊢ 𝐷 = (mDV‘𝑇)
mclspps.e	⊢ 𝐸 = (mEx‘𝑇)
mclspps.c	⊢ 𝐶 = (mCls‘𝑇)
mclspps.1	⊢ (𝜑 → 𝑇 ∈ mFS)
mclspps.2	⊢ (𝜑 → 𝐾 ⊆ 𝐷)
mclspps.3	⊢ (𝜑 → 𝐵 ⊆ 𝐸)
mclspps.j	⊢ 𝐽 = (mPPSt‘𝑇)
mclspps.l	⊢ 𝐿 = (mSubst‘𝑇)
mclspps.v	⊢ 𝑉 = (mVR‘𝑇)
mclspps.h	⊢ 𝐻 = (mVH‘𝑇)
mclspps.w	⊢ 𝑊 = (mVars‘𝑇)
mclspps.4	⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)
mclspps.5	⊢ (𝜑 → 𝑆 ∈ ran 𝐿)
mclspps.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
mclspps.7	⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
mclspps.8	⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)

Assertion

Ref	Expression
mclspps	⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))

Distinct variable groups:   𝑣,𝐸   𝑎,𝑏,𝑣,𝑥,𝑦,𝐻   𝑣,𝑉   𝐾,𝑎,𝑏,𝑣,𝑥,𝑦   𝑇,𝑎,𝑏,𝑣,𝑥,𝑦   𝐿,𝑎,𝑏,𝑣,𝑥,𝑦   𝑆,𝑎,𝑏,𝑣,𝑥,𝑦   𝐵,𝑎,𝑏,𝑣,𝑥,𝑦   𝑊,𝑎,𝑏,𝑣,𝑥,𝑦   𝐶,𝑎,𝑏,𝑣,𝑥,𝑦   𝑀,𝑎,𝑏,𝑣,𝑥,𝑦   𝑣,𝑂,𝑥   𝜑,𝑎,𝑏,𝑣,𝑥,𝑦

Allowed substitution hints:   𝐷(𝑥, 𝑦, 𝑣, 𝑎, 𝑏)   𝑃(𝑥, 𝑦, 𝑣, 𝑎, 𝑏)   𝐸(𝑥, 𝑦, 𝑎, 𝑏)   𝐽(𝑥, 𝑦, 𝑣, 𝑎, 𝑏)   𝑂(𝑦, 𝑎, 𝑏)   𝑉(𝑥, 𝑦, 𝑎, 𝑏)

Proof of Theorem mclspps

Dummy variables 𝑚 𝑜 𝑝 𝑠 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mclspps.5	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)
2		mclspps.l	. . . . 5 ⊢ 𝐿 = (mSubst‘𝑇)
3		mclspps.e	. . . . 5 ⊢ 𝐸 = (mEx‘𝑇)
4	2, 3	msubf 36384	. . . 4 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
5	1, 4	syl 18	. . 3 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
6	5	ffnd 6711	. 2 ⊢ (𝜑 → 𝑆 Fn 𝐸)
7		mclspps.d	. . . 4 ⊢ 𝐷 = (mDV‘𝑇)
8		mclspps.c	. . . 4 ⊢ 𝐶 = (mCls‘𝑇)
9		mclspps.1	. . . 4 ⊢ (𝜑 → 𝑇 ∈ mFS)
10		eqid 2760	. . . . . . . . 9 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
11		mclspps.j	. . . . . . . . 9 ⊢ 𝐽 = (mPPSt‘𝑇)
12	10, 11	mppspst 36426	. . . . . . . 8 ⊢ 𝐽 ⊆ (mPreSt‘𝑇)
13		mclspps.4	. . . . . . . 8 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)
14	12, 13	sselid 3906	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
15	7, 3, 10	elmpst 36388	. . . . . . 7 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀) ∧ (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin) ∧ 𝑃 ∈ 𝐸))
16	14, 15	sylib 221	. . . . . 6 ⊢ (𝜑 → ((𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀) ∧ (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin) ∧ 𝑃 ∈ 𝐸))
17	16	simp1d 1160	. . . . 5 ⊢ (𝜑 → (𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀))
18	17	simpld 500	. . . 4 ⊢ (𝜑 → 𝑀 ⊆ 𝐷)
19	16	simp2d 1161	. . . . 5 ⊢ (𝜑 → (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin))
20	19	simpld 500	. . . 4 ⊢ (𝜑 → 𝑂 ⊆ 𝐸)
21		eqid 2760	. . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
22		mclspps.v	. . . 4 ⊢ 𝑉 = (mVR‘𝑇)
23		mclspps.h	. . . 4 ⊢ 𝐻 = (mVH‘𝑇)
24		mclspps.w	. . . 4 ⊢ 𝑊 = (mVars‘𝑇)
25		mclspps.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
26	25	ralrimiva 3154	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
27	5	ffund 6715	. . . . . 6 ⊢ (𝜑 → Fun 𝑆)
28	5	fdmd 6721	. . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐸)
29	20, 28	sseqtrrd 3945	. . . . . 6 ⊢ (𝜑 → 𝑂 ⊆ dom 𝑆)
30		funimass5 7055	. . . . . 6 ⊢ ((Fun 𝑆 ∧ 𝑂 ⊆ dom 𝑆) → (𝑂 ⊆ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ∀𝑥 ∈ 𝑂 (𝑆‘𝑥) ∈ (𝐾𝐶𝐵)))
31	27, 29, 30	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑂 ⊆ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ∀𝑥 ∈ 𝑂 (𝑆‘𝑥) ∈ (𝐾𝐶𝐵)))
32	26, 31	mpbird 260	. . . 4 ⊢ (𝜑 → 𝑂 ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
33	22, 3, 23	mvhf 36410	. . . . . . 7 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
34	9, 33	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
35	34	ffvelcdmda 7085	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝐸)
36		mclspps.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
37		elpreima 7058	. . . . . . 7 ⊢ (𝑆 Fn 𝐸 → ((𝐻‘𝑣) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑣) ∈ 𝐸 ∧ (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))))
38	6, 37	syl 18	. . . . . 6 ⊢ (𝜑 → ((𝐻‘𝑣) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑣) ∈ 𝐸 ∧ (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))))
39	38	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ((𝐻‘𝑣) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑣) ∈ 𝐸 ∧ (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))))
40	35, 36, 39	mpbir2and 726	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ (◡𝑆 “ (𝐾𝐶𝐵)))
41	9	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → 𝑇 ∈ mFS)
42		mclspps.2	. . . . . 6 ⊢ (𝜑 → 𝐾 ⊆ 𝐷)
43	42	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → 𝐾 ⊆ 𝐷)
44		mclspps.3	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
45	44	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → 𝐵 ⊆ 𝐸)
46	13	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)
47	1	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → 𝑆 ∈ ran 𝐿)
48	25	3ad2antl1 1204	. . . . 5 ⊢ (((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
49	36	3ad2antl1 1204	. . . . 5 ⊢ (((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
50		mclspps.8	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)
51	50	3ad2antl1 1204	. . . . 5 ⊢ (((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)
52		simp21 1225	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))
53		simp22 1226	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → 𝑠 ∈ ran 𝐿)
54		simp23 1227	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
55		simp3 1156	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀))
56	7, 3, 8, 41, 43, 45, 11, 2, 22, 23, 24, 46, 47, 48, 49, 51, 52, 53, 54, 55	mclsppslem 36435	. . . 4 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → (𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)))
57	7, 3, 8, 9, 18, 20, 21, 2, 22, 23, 24, 32, 40, 56	mclsind 36422	. . 3 ⊢ (𝜑 → (𝑀𝐶𝑂) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
58	10, 11, 8	elmpps 36425	. . . . 5 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽 ↔ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ∧ 𝑃 ∈ (𝑀𝐶𝑂)))
59	58	simprbi 503	. . . 4 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽 → 𝑃 ∈ (𝑀𝐶𝑂))
60	13, 59	syl 18	. . 3 ⊢ (𝜑 → 𝑃 ∈ (𝑀𝐶𝑂))
61	57, 60	sseldd 3909	. 2 ⊢ (𝜑 → 𝑃 ∈ (◡𝑆 “ (𝐾𝐶𝐵)))
62		elpreima 7058	. . 3 ⊢ (𝑆 Fn 𝐸 → (𝑃 ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑃 ∈ 𝐸 ∧ (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))))
63	62	simplbda 505	. 2 ⊢ ((𝑆 Fn 𝐸 ∧ 𝑃 ∈ (◡𝑆 “ (𝐾𝐶𝐵))) → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))
64	6, 61, 63	syl2anc 596	1 ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∪ cun 4149  ⟨cotp 4857   class class class wbr 5102   × cxp 5648  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   “ cima 5653  Fun wfun 6533   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421  Fincfn 8977  mVRcmvar 36313  mAxcmax 36317  mExcmex 36319  mDVcmdv 36320  mVarscmvrs 36321  mSubstcmsub 36323  mVHcmvh 36324  mPreStcmpst 36325  mFScmfs 36328  mClscmcls 36329  mPPStcmpps 36330

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-er 8721  df-map 8853  df-pm 8854  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-card 10024  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-nn 12340  df-2 12409  df-n0 12611  df-xnn0 12684  df-z 12698  df-uz 12970  df-fz 13644  df-fzo 13793  df-seq 14149  df-hash 14479  df-word 14663  df-lsw 14712  df-concat 14720  df-s1 14747  df-substr 14794  df-pfx 14826  df-struct 17331  df-sets 17348  df-slot 17366  df-ndx 17378  df-base 17394  df-ress 17415  df-plusg 17447  df-0g 17618  df-gsum 17619  df-mgm 18822  df-sgrp 18914  df-mnd 18930  df-mhm 18984  df-submnd 18985  df-frmd 19051  df-vrmd 19052  df-mrex 36338  df-mex 36339  df-mdv 36340  df-mvrs 36341  df-mrsub 36342  df-msub 36343  df-mvh 36344  df-mpst 36345  df-msr 36346  df-msta 36347  df-mfs 36348  df-mcls 36349  df-mpps 36350

This theorem is used by: (None)