Theorem brwdom2 8478

Description: Alternate characterization of the weak dominance predicate which does not require special treatment of the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)

Assertion

Ref	Expression
brwdom2	⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))

Distinct variable groups:   𝑦,𝑋,𝑧   𝑦,𝑌,𝑧

Allowed substitution hints:   𝑉(𝑦,𝑧)

Proof of Theorem brwdom2

Dummy variables 𝑥 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ V)
2		0wdom 8475	. . . . . 6 ⊢ (𝑌 ∈ V → ∅ ≼* 𝑌)
3		breq1 4656	. . . . . 6 ⊢ (𝑋 = ∅ → (𝑋 ≼* 𝑌 ↔ ∅ ≼* 𝑌))
4	2, 3	syl5ibrcom 237	. . . . 5 ⊢ (𝑌 ∈ V → (𝑋 = ∅ → 𝑋 ≼* 𝑌))
5	4	imp 445	. . . 4 ⊢ ((𝑌 ∈ V ∧ 𝑋 = ∅) → 𝑋 ≼* 𝑌)
6		0elpw 4834	. . . . . . 7 ⊢ ∅ ∈ 𝒫 𝑌
7		f1o0 6173	. . . . . . . 8 ⊢ ∅:∅–1-1-onto→∅
8		f1ofo 6144	. . . . . . . 8 ⊢ (∅:∅–1-1-onto→∅ → ∅:∅–onto→∅)
9		0ex 4790	. . . . . . . . 9 ⊢ ∅ ∈ V
10		foeq1 6111	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑧:∅–onto→∅ ↔ ∅:∅–onto→∅))
11	9, 10	spcev 3300	. . . . . . . 8 ⊢ (∅:∅–onto→∅ → ∃𝑧 𝑧:∅–onto→∅)
12	7, 8, 11	mp2b 10	. . . . . . 7 ⊢ ∃𝑧 𝑧:∅–onto→∅
13		foeq2 6112	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑧:𝑦–onto→∅ ↔ 𝑧:∅–onto→∅))
14	13	exbidv 1850	. . . . . . . 8 ⊢ (𝑦 = ∅ → (∃𝑧 𝑧:𝑦–onto→∅ ↔ ∃𝑧 𝑧:∅–onto→∅))
15	14	rspcev 3309	. . . . . . 7 ⊢ ((∅ ∈ 𝒫 𝑌 ∧ ∃𝑧 𝑧:∅–onto→∅) → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→∅)
16	6, 12, 15	mp2an 708	. . . . . 6 ⊢ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→∅
17		foeq3 6113	. . . . . . . 8 ⊢ (𝑋 = ∅ → (𝑧:𝑦–onto→𝑋 ↔ 𝑧:𝑦–onto→∅))
18	17	exbidv 1850	. . . . . . 7 ⊢ (𝑋 = ∅ → (∃𝑧 𝑧:𝑦–onto→𝑋 ↔ ∃𝑧 𝑧:𝑦–onto→∅))
19	18	rexbidv 3052	. . . . . 6 ⊢ (𝑋 = ∅ → (∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→∅))
20	16, 19	mpbiri 248	. . . . 5 ⊢ (𝑋 = ∅ → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋)
21	20	adantl 482	. . . 4 ⊢ ((𝑌 ∈ V ∧ 𝑋 = ∅) → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋)
22	5, 21	2thd 255	. . 3 ⊢ ((𝑌 ∈ V ∧ 𝑋 = ∅) → (𝑋 ≼* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))
23		brwdomn0 8474	. . . . 5 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑥 𝑥:𝑌–onto→𝑋))
24	23	adantl 482	. . . 4 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑋 ≼* 𝑌 ↔ ∃𝑥 𝑥:𝑌–onto→𝑋))
25		foeq1 6111	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥:𝑌–onto→𝑋 ↔ 𝑧:𝑌–onto→𝑋))
26	25	cbvexv 2275	. . . . . 6 ⊢ (∃𝑥 𝑥:𝑌–onto→𝑋 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)
27		pwidg 4173	. . . . . . . . 9 ⊢ (𝑌 ∈ V → 𝑌 ∈ 𝒫 𝑌)
28	27	ad2antrr 762	. . . . . . . 8 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ ∃𝑧 𝑧:𝑌–onto→𝑋) → 𝑌 ∈ 𝒫 𝑌)
29		foeq2 6112	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑧:𝑦–onto→𝑋 ↔ 𝑧:𝑌–onto→𝑋))
30	29	exbidv 1850	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦–onto→𝑋 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
31	30	rspcev 3309	. . . . . . . 8 ⊢ ((𝑌 ∈ 𝒫 𝑌 ∧ ∃𝑧 𝑧:𝑌–onto→𝑋) → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋)
32	28, 31	sylancom 701	. . . . . . 7 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ ∃𝑧 𝑧:𝑌–onto→𝑋) → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋)
33	32	ex 450	. . . . . 6 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑧 𝑧:𝑌–onto→𝑋 → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))
34	26, 33	syl5bi 232	. . . . 5 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑥 𝑥:𝑌–onto→𝑋 → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))
35		n0 3931	. . . . . . . . . . 11 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝑋)
36	35	biimpi 206	. . . . . . . . . 10 ⊢ (𝑋 ≠ ∅ → ∃𝑤 𝑤 ∈ 𝑋)
37	36	ad2antlr 763	. . . . . . . . 9 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) → ∃𝑤 𝑤 ∈ 𝑋)
38		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
39		difexg 4808	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ V → (𝑌 ∖ 𝑦) ∈ V)
40		snex 4908	. . . . . . . . . . . . . 14 ⊢ {𝑤} ∈ V
41		xpexg 6960	. . . . . . . . . . . . . 14 ⊢ (((𝑌 ∖ 𝑦) ∈ V ∧ {𝑤} ∈ V) → ((𝑌 ∖ 𝑦) × {𝑤}) ∈ V)
42	39, 40, 41	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ V → ((𝑌 ∖ 𝑦) × {𝑤}) ∈ V)
43		unexg 6959	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ V ∧ ((𝑌 ∖ 𝑦) × {𝑤}) ∈ V) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) ∈ V)
44	38, 42, 43	sylancr 695	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ V → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) ∈ V)
45	44	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) ∈ V)
46	45	ad2antrr 762	. . . . . . . . . 10 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) ∈ V)
47		fofn 6117	. . . . . . . . . . . . . . 15 ⊢ (𝑧:𝑦–onto→𝑋 → 𝑧 Fn 𝑦)
48	47	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋) → 𝑧 Fn 𝑦)
49	48	ad2antlr 763	. . . . . . . . . . . . 13 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → 𝑧 Fn 𝑦)
50		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
51		fnconstg 6093	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ V → ((𝑌 ∖ 𝑦) × {𝑤}) Fn (𝑌 ∖ 𝑦))
52	50, 51	mp1i 13	. . . . . . . . . . . . 13 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → ((𝑌 ∖ 𝑦) × {𝑤}) Fn (𝑌 ∖ 𝑦))
53		disjdif 4040	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∩ (𝑌 ∖ 𝑦)) = ∅
54	53	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑦 ∩ (𝑌 ∖ 𝑦)) = ∅)
55		fnun 5997	. . . . . . . . . . . . 13 ⊢ (((𝑧 Fn 𝑦 ∧ ((𝑌 ∖ 𝑦) × {𝑤}) Fn (𝑌 ∖ 𝑦)) ∧ (𝑦 ∩ (𝑌 ∖ 𝑦)) = ∅) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌 ∖ 𝑦)))
56	49, 52, 54, 55	syl21anc 1325	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌 ∖ 𝑦)))
57		elpwi 4168	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝒫 𝑌 → 𝑦 ⊆ 𝑌)
58		undif 4049	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ 𝑌 ↔ (𝑦 ∪ (𝑌 ∖ 𝑦)) = 𝑌)
59	57, 58	sylib 208	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝒫 𝑌 → (𝑦 ∪ (𝑌 ∖ 𝑦)) = 𝑌)
60	59	ad2antrl 764	. . . . . . . . . . . . . 14 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) → (𝑦 ∪ (𝑌 ∖ 𝑦)) = 𝑌)
61	60	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑦 ∪ (𝑌 ∖ 𝑦)) = 𝑌)
62	61	fneq2d 5982	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → ((𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌 ∖ 𝑦)) ↔ (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) Fn 𝑌))
63	56, 62	mpbid 222	. . . . . . . . . . 11 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) Fn 𝑌)
64		rnun 5541	. . . . . . . . . . . 12 ⊢ ran (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) = (ran 𝑧 ∪ ran ((𝑌 ∖ 𝑦) × {𝑤}))
65		forn 6118	. . . . . . . . . . . . . . . 16 ⊢ (𝑧:𝑦–onto→𝑋 → ran 𝑧 = 𝑋)
66	65	ad2antll 765	. . . . . . . . . . . . . . 15 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) → ran 𝑧 = 𝑋)
67	66	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → ran 𝑧 = 𝑋)
68	67	uneq1d 3766	. . . . . . . . . . . . 13 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (ran 𝑧 ∪ ran ((𝑌 ∖ 𝑦) × {𝑤})) = (𝑋 ∪ ran ((𝑌 ∖ 𝑦) × {𝑤})))
69		fconst6g 6094	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ 𝑋 → ((𝑌 ∖ 𝑦) × {𝑤}):(𝑌 ∖ 𝑦)⟶𝑋)
70		frn 6053	. . . . . . . . . . . . . . . 16 ⊢ (((𝑌 ∖ 𝑦) × {𝑤}):(𝑌 ∖ 𝑦)⟶𝑋 → ran ((𝑌 ∖ 𝑦) × {𝑤}) ⊆ 𝑋)
71	69, 70	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝑋 → ran ((𝑌 ∖ 𝑦) × {𝑤}) ⊆ 𝑋)
72	71	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → ran ((𝑌 ∖ 𝑦) × {𝑤}) ⊆ 𝑋)
73		ssequn2 3786	. . . . . . . . . . . . . 14 ⊢ (ran ((𝑌 ∖ 𝑦) × {𝑤}) ⊆ 𝑋 ↔ (𝑋 ∪ ran ((𝑌 ∖ 𝑦) × {𝑤})) = 𝑋)
74	72, 73	sylib 208	. . . . . . . . . . . . 13 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑋 ∪ ran ((𝑌 ∖ 𝑦) × {𝑤})) = 𝑋)
75	68, 74	eqtrd 2656	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (ran 𝑧 ∪ ran ((𝑌 ∖ 𝑦) × {𝑤})) = 𝑋)
76	64, 75	syl5eq 2668	. . . . . . . . . . 11 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → ran (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) = 𝑋)
77		df-fo 5894	. . . . . . . . . . 11 ⊢ ((𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})):𝑌–onto→𝑋 ↔ ((𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) Fn 𝑌 ∧ ran (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) = 𝑋))
78	63, 76, 77	sylanbrc 698	. . . . . . . . . 10 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})):𝑌–onto→𝑋)
79		foeq1 6111	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) → (𝑥:𝑌–onto→𝑋 ↔ (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})):𝑌–onto→𝑋))
80	79	spcegv 3294	. . . . . . . . . 10 ⊢ ((𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) ∈ V → ((𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})):𝑌–onto→𝑋 → ∃𝑥 𝑥:𝑌–onto→𝑋))
81	46, 78, 80	sylc 65	. . . . . . . . 9 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → ∃𝑥 𝑥:𝑌–onto→𝑋)
82	37, 81	exlimddv 1863	. . . . . . . 8 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) → ∃𝑥 𝑥:𝑌–onto→𝑋)
83	82	expr 643	. . . . . . 7 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝑧:𝑦–onto→𝑋 → ∃𝑥 𝑥:𝑌–onto→𝑋))
84	83	exlimdv 1861	. . . . . 6 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ 𝑦 ∈ 𝒫 𝑌) → (∃𝑧 𝑧:𝑦–onto→𝑋 → ∃𝑥 𝑥:𝑌–onto→𝑋))
85	84	rexlimdva 3031	. . . . 5 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋 → ∃𝑥 𝑥:𝑌–onto→𝑋))
86	34, 85	impbid 202	. . . 4 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑥 𝑥:𝑌–onto→𝑋 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))
87	24, 86	bitrd 268	. . 3 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑋 ≼* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))
88	22, 87	pm2.61dane 2881	. 2 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))
89	1, 88	syl 17	1 ⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177   class class class wbr 4653   × cxp 5112  ran crn 5115   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887   ≼^* cwdom 8462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-wdom 8464

This theorem is referenced by:  brwdom3  8487