Theorem fodomr 9167

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 8990	. . . 4 ⊢ Rel ≼
2	1	brrelex2i 5746	. . 3 ⊢ (𝐵 ≼ 𝐴 → 𝐴 ∈ V)
3	2	adantl 481	. 2 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ∈ V)
4	1	brrelex1i 5745	. . . 4 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V)
5		0sdomg 9143	. . . . 5 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ 𝐵 ≠ ∅))
6		n0 4359	. . . . 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 8998	. . 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 5335	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 ∖ ran 𝑔) ∈ V)
13		vsnex 5440	. . . . . . . . . 10 ⊢ {𝑧} ∈ V
14		xpexg 7769	. . . . . . . . . 10 ⊢ (((𝐴 ∖ ran 𝑔) ∈ V ∧ {𝑧} ∈ V) → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V)
15	12, 13, 14	sylancl 586	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V)
16		vex 3482	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
17	16	cnvex 7948	. . . . . . . . 9 ⊢ ◡𝑔 ∈ V
18	15, 17	jctil 519	. . . . . . . 8 ⊢ (𝐴 ∈ V → (◡𝑔 ∈ V ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V))
19		unexb 7766	. . . . . . . 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 6568	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐵–1-1→𝐴 ↔ (𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔))
22	21	simprbi 496	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → Fun ◡𝑔)
23		vex 3482	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
24	23	fconst 6795	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧}
25		ffun 6740	. . . . . . . . . . . . 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 5700	. . . . . . . . . . . . . 14 ⊢ ran 𝑔 = dom ◡𝑔
29	28	eqcomi 2744	. . . . . . . . . . . . 13 ⊢ dom ◡𝑔 = ran 𝑔
30	23	snnz 4781	. . . . . . . . . . . . . 14 ⊢ {𝑧} ≠ ∅
31		dmxp 5942	. . . . . . . . . . . . . 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 4226	. . . . . . . . . . . 12 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔))
34		disjdif 4478	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔)) = ∅
35	33, 34	eqtri 2763	. . . . . . . . . . 11 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅
36		funun 6614	. . . . . . . . . . 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 586	. . . . . . . . . 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 5924	. . . . . . . . . . . 12 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
40	28	uneq1i 4174	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
41	32	uneq2i 4175	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
42	39, 40, 41	3eqtr2i 2769	. . . . . . . . . . 11 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
43		f1f 6805	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐵–1-1→𝐴 → 𝑔:𝐵⟶𝐴)
44	43	frnd 6745	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → ran 𝑔 ⊆ 𝐴)
45		undif 4488	. . . . . . . . . . . 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 2787	. . . . . . . . . 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 6566	. . . . . . . . 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 583	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴)
51		rnun 6168	. . . . . . . . 9 ⊢ ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧}))
52		dfdm4 5909	. . . . . . . . . . . 12 ⊢ dom 𝑔 = ran ◡𝑔
53		f1dm 6809	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → dom 𝑔 = 𝐵)
54	52, 53	eqtr3id 2789	. . . . . . . . . . 11 ⊢ (𝑔:𝐵–1-1→𝐴 → ran ◡𝑔 = 𝐵)
55	54	uneq1d 4177	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})))
56		xpeq1 5703	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
57		0xp 5787	. . . . . . . . . . . . . . . . 17 ⊢ (∅ × {𝑧}) = ∅
58	56, 57	eqtrdi 2791	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
59	58	rneqd 5952	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ran ∅)
60		rn0 5939	. . . . . . . . . . . . . . 15 ⊢ ran ∅ = ∅
61	59, 60	eqtrdi 2791	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
62		0ss 4406	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ 𝐵
63	61, 62	eqsstrdi 4050	. . . . . . . . . . . . 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 6192	. . . . . . . . . . . . . . 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 4813	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐵 → {𝑧} ⊆ 𝐵)
68	67	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → {𝑧} ⊆ 𝐵)
69	66, 68	eqsstrd 4034	. . . . . . . . . . . . 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 3023	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
72		ssequn2 4199	. . . . . . . . . . 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 2797	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
75	51, 74	eqtrid 2787	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
76		df-fo 6569	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵)
78		foeq1 6817	. . . . . . . 8 ⊢ (𝑓 = (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝐴–onto→𝐵 ↔ (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵))
79	78	spcegv 3597	. . . . . . 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 1931	. . . 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 1931	. 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 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  Vcvv 3478   ∖ cdif 3960   ∪ cun 3961   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  {csn 4631   class class class wbr 5148   × cxp 5687  ^◡ccnv 5688  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  ⟶wf 6559  –1-1→wf1 6560  –onto→wfo 6561   ≼ cdom 8982   ≺ csdm 8983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-en 8985  df-dom 8986  df-sdom 8987

This theorem is referenced by:  pwdom  9168  fodomfibOLD  9369  domwdom  9612  iunfictbso  10152  fodomb  10564  brdom3  10566  konigthlem  10606  1stcfb  23469  ovoliunnul  25556  sigapildsys  34143  carsgclctunlem3  34302  ovoliunnfl  37649  voliunnfl  37651  volsupnfl  37652  modelaxreplem1  44943  nnfoctb  44987  caragenunicl  46480