Theorem fodomr 8111

Description: There exists a mapping from a set onto any (nonempty) set that it dominates. (Contributed by NM, 23-Mar-2006.)

Assertion

Ref	Expression
fodomr	⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem fodomr

Dummy variables 𝑔 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldom 7961	. . . 4 ⊢ Rel ≼
2	1	brrelex2i 5159	. . 3 ⊢ (𝐵 ≼ 𝐴 → 𝐴 ∈ V)
3	2	adantl 482	. 2 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ∈ V)
4	1	brrelexi 5158	. . . 4 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V)
5		0sdomg 8089	. . . . 5 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ 𝐵 ≠ ∅))
6		n0 3931	. . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐵)
7	5, 6	syl6bb 276	. . . 4 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵))
8	4, 7	syl 17	. . 3 ⊢ (𝐵 ≼ 𝐴 → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵))
9	8	biimpac 503	. 2 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑧 𝑧 ∈ 𝐵)
10		brdomi 7966	. . 3 ⊢ (𝐵 ≼ 𝐴 → ∃𝑔 𝑔:𝐵–1-1→𝐴)
11	10	adantl 482	. 2 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑔 𝑔:𝐵–1-1→𝐴)
12		difexg 4808	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 ∖ ran 𝑔) ∈ V)
13		snex 4908	. . . . . . . . . 10 ⊢ {𝑧} ∈ V
14		xpexg 6960	. . . . . . . . . 10 ⊢ (((𝐴 ∖ ran 𝑔) ∈ V ∧ {𝑧} ∈ V) → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V)
15	12, 13, 14	sylancl 694	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V)
16		vex 3203	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
17	16	cnvex 7113	. . . . . . . . 9 ⊢ ◡𝑔 ∈ V
18	15, 17	jctil 560	. . . . . . . 8 ⊢ (𝐴 ∈ V → (◡𝑔 ∈ V ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V))
19		unexb 6958	. . . . . . . 8 ⊢ ((◡𝑔 ∈ V ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V) ↔ (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V)
20	18, 19	sylib 208	. . . . . . 7 ⊢ (𝐴 ∈ V → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V)
21		df-f1 5893	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐵–1-1→𝐴 ↔ (𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔))
22	21	simprbi 480	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → Fun ◡𝑔)
23		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
24	23	fconst 6091	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧}
25		ffun 6048	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧} → Fun ((𝐴 ∖ ran 𝑔) × {𝑧}))
26	24, 25	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})
27	22, 26	jctir 561	. . . . . . . . . . 11 ⊢ (𝑔:𝐵–1-1→𝐴 → (Fun ◡𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})))
28		df-rn 5125	. . . . . . . . . . . . . 14 ⊢ ran 𝑔 = dom ◡𝑔
29	28	eqcomi 2631	. . . . . . . . . . . . 13 ⊢ dom ◡𝑔 = ran 𝑔
30	23	snnz 4309	. . . . . . . . . . . . . 14 ⊢ {𝑧} ≠ ∅
31		dmxp 5344	. . . . . . . . . . . . . 14 ⊢ ({𝑧} ≠ ∅ → dom ((𝐴 ∖ ran 𝑔) × {𝑧}) = (𝐴 ∖ ran 𝑔))
32	30, 31	ax-mp 5	. . . . . . . . . . . . 13 ⊢ dom ((𝐴 ∖ ran 𝑔) × {𝑧}) = (𝐴 ∖ ran 𝑔)
33	29, 32	ineq12i 3812	. . . . . . . . . . . 12 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔))
34		disjdif 4040	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔)) = ∅
35	33, 34	eqtri 2644	. . . . . . . . . . 11 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅
36		funun 5932	. . . . . . . . . . 11 ⊢ (((Fun ◡𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})) ∧ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅) → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
37	27, 35, 36	sylancl 694	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
38	37	adantl 482	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
39		dmun 5331	. . . . . . . . . . . 12 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
40	28	uneq1i 3763	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
41	32	uneq2i 3764	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
42	39, 40, 41	3eqtr2i 2650	. . . . . . . . . . 11 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
43		f1f 6101	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐵–1-1→𝐴 → 𝑔:𝐵⟶𝐴)
44		frn 6053	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐵⟶𝐴 → ran 𝑔 ⊆ 𝐴)
45	43, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → ran 𝑔 ⊆ 𝐴)
46		undif 4049	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ⊆ 𝐴 ↔ (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
47	45, 46	sylib 208	. . . . . . . . . . 11 ⊢ (𝑔:𝐵–1-1→𝐴 → (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
48	42, 47	syl5eq 2668	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
49	48	adantl 482	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
50		df-fn 5891	. . . . . . . . 9 ⊢ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ↔ (Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∧ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴))
51	38, 49, 50	sylanbrc 698	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴)
52		rnun 5541	. . . . . . . . 9 ⊢ ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧}))
53		dfdm4 5316	. . . . . . . . . . . 12 ⊢ dom 𝑔 = ran ◡𝑔
54		f1dm 6105	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → dom 𝑔 = 𝐵)
55	53, 54	syl5eqr 2670	. . . . . . . . . . 11 ⊢ (𝑔:𝐵–1-1→𝐴 → ran ◡𝑔 = 𝐵)
56	55	uneq1d 3766	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})))
57		xpeq1 5128	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
58		0xp 5199	. . . . . . . . . . . . . . . . 17 ⊢ (∅ × {𝑧}) = ∅
59	57, 58	syl6eq 2672	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
60	59	rneqd 5353	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ran ∅)
61		rn0 5377	. . . . . . . . . . . . . . 15 ⊢ ran ∅ = ∅
62	60, 61	syl6eq 2672	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
63		0ss 3972	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ 𝐵
64	62, 63	syl6eqss 3655	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
65	64	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
66		rnxp 5564	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
67	66	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
68		snssi 4339	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐵 → {𝑧} ⊆ 𝐵)
69	68	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → {𝑧} ⊆ 𝐵)
70	67, 69	eqsstrd 3639	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
71	70	ex 450	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
72	65, 71	pm2.61ine 2877	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
73		ssequn2 3786	. . . . . . . . . . 11 ⊢ (ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵 ↔ (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
74	72, 73	sylib 208	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐵 → (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
75	56, 74	sylan9eqr 2678	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
76	52, 75	syl5eq 2668	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
77		df-fo 5894	. . . . . . . 8 ⊢ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵 ↔ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ∧ ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵))
78	51, 76, 77	sylanbrc 698	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵)
79		foeq1 6111	. . . . . . . 8 ⊢ (𝑓 = (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝐴–onto→𝐵 ↔ (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵))
80	79	spcegv 3294	. . . . . . 7 ⊢ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V → ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵))
81	20, 78, 80	syl2im 40	. . . . . 6 ⊢ (𝐴 ∈ V → ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵))
82	81	expdimp 453	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝐵) → (𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵))
83	82	exlimdv 1861	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝐵) → (∃𝑔 𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵))
84	83	ex 450	. . 3 ⊢ (𝐴 ∈ V → (𝑧 ∈ 𝐵 → (∃𝑔 𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
85	84	exlimdv 1861	. 2 ⊢ (𝐴 ∈ V → (∃𝑧 𝑧 ∈ 𝐵 → (∃𝑔 𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
86	3, 9, 11, 85	syl3c 66	1 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886   ≼ cdom 7953   ≺ csdm 7954

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-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by:  pwdom  8112  fodomfib  8240  domwdom  8479  iunfictbso  8937  fodomb  9348  brdom3  9350  konigthlem  9390  1stcfb  21248  ovoliunnul  23275  sigapildsys  30225  carsgclctunlem3  30382  ovoliunnfl  33451  voliunnfl  33453  volsupnfl  33454  nnfoctb  39213  caragenunicl  40738