Theorem fodomr 9068

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 8901	. . . 4 ⊢ Rel ≼
2	1	brrelex2i 5689	. . 3 ⊢ (𝐵 ≼ 𝐴 → 𝐴 ∈ V)
3	2	adantl 481	. 2 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ∈ V)
4	1	brrelex1i 5688	. . . 4 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V)
5		0sdomg 9046	. . . . 5 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ 𝐵 ≠ ∅))
6		n0 4307	. . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐵)
7	5, 6	bitrdi 287	. . . 4 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵))
8	4, 7	syl 17	. . 3 ⊢ (𝐵 ≼ 𝐴 → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵))
9	8	biimpac 478	. 2 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑧 𝑧 ∈ 𝐵)
10		brdomi 8908	. . 3 ⊢ (𝐵 ≼ 𝐴 → ∃𝑔 𝑔:𝐵–1-1→𝐴)
11	10	adantl 481	. 2 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑔 𝑔:𝐵–1-1→𝐴)
12		difexg 5276	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 ∖ ran 𝑔) ∈ V)
13		vsnex 5381	. . . . . . . . . 10 ⊢ {𝑧} ∈ V
14		xpexg 7705	. . . . . . . . . 10 ⊢ (((𝐴 ∖ ran 𝑔) ∈ V ∧ {𝑧} ∈ V) → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V)
15	12, 13, 14	sylancl 587	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V)
16		vex 3446	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
17	16	cnvex 7877	. . . . . . . . 9 ⊢ ◡𝑔 ∈ V
18	15, 17	jctil 519	. . . . . . . 8 ⊢ (𝐴 ∈ V → (◡𝑔 ∈ V ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V))
19		unexb 7702	. . . . . . . 8 ⊢ ((◡𝑔 ∈ V ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V) ↔ (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V)
20	18, 19	sylib 218	. . . . . . 7 ⊢ (𝐴 ∈ V → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V)
21		df-f1 6505	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐵–1-1→𝐴 ↔ (𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔))
22	21	simprbi 497	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → Fun ◡𝑔)
23		vex 3446	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
24	23	fconst 6728	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧}
25		ffun 6673	. . . . . . . . . . . . 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 520	. . . . . . . . . . 11 ⊢ (𝑔:𝐵–1-1→𝐴 → (Fun ◡𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})))
28		df-rn 5643	. . . . . . . . . . . . . 14 ⊢ ran 𝑔 = dom ◡𝑔
29	28	eqcomi 2746	. . . . . . . . . . . . 13 ⊢ dom ◡𝑔 = ran 𝑔
30	23	snnz 4735	. . . . . . . . . . . . . 14 ⊢ {𝑧} ≠ ∅
31		dmxp 5886	. . . . . . . . . . . . . 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 4172	. . . . . . . . . . . 12 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔))
34		disjdif 4426	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔)) = ∅
35	33, 34	eqtri 2760	. . . . . . . . . . 11 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅
36		funun 6546	. . . . . . . . . . 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 587	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
38	37	adantl 481	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
39		dmun 5867	. . . . . . . . . . . 12 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
40	28	uneq1i 4118	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
41	32	uneq2i 4119	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
42	39, 40, 41	3eqtr2i 2766	. . . . . . . . . . 11 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
43		f1f 6738	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐵–1-1→𝐴 → 𝑔:𝐵⟶𝐴)
44	43	frnd 6678	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → ran 𝑔 ⊆ 𝐴)
45		undif 4436	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ⊆ 𝐴 ↔ (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
46	44, 45	sylib 218	. . . . . . . . . . 11 ⊢ (𝑔:𝐵–1-1→𝐴 → (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
47	42, 46	eqtrid 2784	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
48	47	adantl 481	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
49		df-fn 6503	. . . . . . . . 9 ⊢ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ↔ (Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∧ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴))
50	38, 48, 49	sylanbrc 584	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴)
51		rnun 6111	. . . . . . . . 9 ⊢ ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧}))
52		dfdm4 5852	. . . . . . . . . . . 12 ⊢ dom 𝑔 = ran ◡𝑔
53		f1dm 6742	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → dom 𝑔 = 𝐵)
54	52, 53	eqtr3id 2786	. . . . . . . . . . 11 ⊢ (𝑔:𝐵–1-1→𝐴 → ran ◡𝑔 = 𝐵)
55	54	uneq1d 4121	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})))
56		xpeq1 5646	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
57		0xp 5731	. . . . . . . . . . . . . . . . 17 ⊢ (∅ × {𝑧}) = ∅
58	56, 57	eqtrdi 2788	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
59	58	rneqd 5895	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ran ∅)
60		rn0 5883	. . . . . . . . . . . . . . 15 ⊢ ran ∅ = ∅
61	59, 60	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
62		0ss 4354	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ 𝐵
63	61, 62	eqsstrdi 3980	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
64	63	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
65		rnxp 6136	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
66	65	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
67		snssi 4766	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐵 → {𝑧} ⊆ 𝐵)
68	67	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → {𝑧} ⊆ 𝐵)
69	66, 68	eqsstrd 3970	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
70	69	ex 412	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
71	64, 70	pm2.61ine 3016	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
72		ssequn2 4143	. . . . . . . . . . 11 ⊢ (ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵 ↔ (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
73	71, 72	sylib 218	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐵 → (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
74	55, 73	sylan9eqr 2794	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
75	51, 74	eqtrid 2784	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
76		df-fo 6506	. . . . . . . 8 ⊢ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵 ↔ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ∧ ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵))
77	50, 75, 76	sylanbrc 584	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵)
78		foeq1 6750	. . . . . . . 8 ⊢ (𝑓 = (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝐴–onto→𝐵 ↔ (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵))
79	78	spcegv 3553	. . . . . . 7 ⊢ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V → ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵))
80	20, 77, 79	syl2im 40	. . . . . 6 ⊢ (𝐴 ∈ V → ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵))
81	80	expdimp 452	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝐵) → (𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵))
82	81	exlimdv 1935	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝐵) → (∃𝑔 𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵))
83	82	ex 412	. . 3 ⊢ (𝐴 ∈ V → (𝑧 ∈ 𝐵 → (∃𝑔 𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
84	83	exlimdv 1935	. 2 ⊢ (𝐴 ∈ V → (∃𝑧 𝑧 ∈ 𝐵 → (∃𝑔 𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
85	3, 9, 11, 84	syl3c 66	1 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  {csn 4582   class class class wbr 5100   × cxp 5630  ^◡ccnv 5631  dom cdm 5632  ran crn 5633  Fun wfun 6494   Fn wfn 6495  ⟶wf 6496  –1-1→wf1 6497  –onto→wfo 6498   ≼ cdom 8893   ≺ csdm 8894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-en 8896  df-dom 8897  df-sdom 8898

This theorem is referenced by:  pwdom  9069  fodomfibOLD  9243  domwdom  9491  iunfictbso  10036  fodomb  10448  brdom3  10450  konigthlem  10491  1stcfb  23401  ovoliunnul  25476  sigapildsys  34340  carsgclctunlem3  34498  ovoliunnfl  37913  voliunnfl  37915  volsupnfl  37916  modelaxreplem1  45334  nnfoctb  45408  caragenunicl  46882