Theorem fodomr 9112

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 8945	. . . 4 ⊢ Rel ≼
2	1	brrelex2i 5716	. . 3 ⊢ (𝐵 ≼ 𝐴 → 𝐴 ∈ V)
3	2	adantl 486	. 2 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ∈ V)
4	1	brrelex1i 5715	. . . 4 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V)
5		0sdomg 9090	. . . . 5 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ 𝐵 ≠ ∅))
6		n0 4314	. . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐵)
7	5, 6	bitrdi 290	. . . 4 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵))
8	4, 7	syl 18	. . 3 ⊢ (𝐵 ≼ 𝐴 → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵))
9	8	biimpac 483	. 2 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑧 𝑧 ∈ 𝐵)
10		brdomi 8952	. . 3 ⊢ (𝐵 ≼ 𝐴 → ∃𝑔 𝑔:𝐵–1-1→𝐴)
11	10	adantl 486	. 2 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑔 𝑔:𝐵–1-1→𝐴)
12		difexg 5297	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 ∖ ran 𝑔) ∈ V)
13		vsnex 5404	. . . . . . . . . 10 ⊢ {𝑧} ∈ V
14		xpexg 7745	. . . . . . . . . 10 ⊢ (((𝐴 ∖ ran 𝑔) ∈ V ∧ {𝑧} ∈ V) → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V)
15	12, 13, 14	sylancl 597	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V)
16		vex 3467	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
17	16	cnvex 7918	. . . . . . . . 9 ⊢ ◡𝑔 ∈ V
18	15, 17	jctil 528	. . . . . . . 8 ⊢ (𝐴 ∈ V → (◡𝑔 ∈ V ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V))
19		unexb 7742	. . . . . . . 8 ⊢ ((◡𝑔 ∈ V ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V) ↔ (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V)
20	18, 19	sylib 221	. . . . . . 7 ⊢ (𝐴 ∈ V → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V)
21		df-f1 6539	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐵–1-1→𝐴 ↔ (𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔))
22	21	simprbi 502	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → Fun ◡𝑔)
23		vex 3467	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
24	23	fconst 6762	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧}
25		ffun 6706	. . . . . . . . . . . . 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 529	. . . . . . . . . . 11 ⊢ (𝑔:𝐵–1-1→𝐴 → (Fun ◡𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})))
28		df-rn 5670	. . . . . . . . . . . . . 14 ⊢ ran 𝑔 = dom ◡𝑔
29	28	eqcomi 2778	. . . . . . . . . . . . 13 ⊢ dom ◡𝑔 = ran 𝑔
30	23	snnz 4744	. . . . . . . . . . . . . 14 ⊢ {𝑧} ≠ ∅
31		dmxp 5917	. . . . . . . . . . . . . 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 4179	. . . . . . . . . . . 12 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔))
34		disjdif 4435	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔)) = ∅
35	33, 34	eqtri 2792	. . . . . . . . . . 11 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅
36		funun 6580	. . . . . . . . . . 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 597	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
38	37	adantl 486	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
39		dmun 5898	. . . . . . . . . . . 12 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
40	28	uneq1i 4126	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
41	32	uneq2i 4127	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
42	39, 40, 41	3eqtr2i 2798	. . . . . . . . . . 11 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
43		f1f 6772	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐵–1-1→𝐴 → 𝑔:𝐵⟶𝐴)
44	43	frnd 6712	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → ran 𝑔 ⊆ 𝐴)
45		undif 4445	. . . . . . . . . . . 12 ⊢ (ran 𝑔 ⊆ 𝐴 ↔ (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
46	44, 45	sylib 221	. . . . . . . . . . 11 ⊢ (𝑔:𝐵–1-1→𝐴 → (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
47	42, 46	eqtrid 2816	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
48	47	adantl 486	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
49		df-fn 6537	. . . . . . . . 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 594	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴)
51		rnun 6140	. . . . . . . . 9 ⊢ ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧}))
52		dfdm4 5883	. . . . . . . . . . . 12 ⊢ dom 𝑔 = ran ◡𝑔
53		f1dm 6778	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → dom 𝑔 = 𝐵)
54	52, 53	eqtr3id 2818	. . . . . . . . . . 11 ⊢ (𝑔:𝐵–1-1→𝐴 → ran ◡𝑔 = 𝐵)
55	54	uneq1d 4129	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})))
56		xpeq1 5673	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
57		0xp 5758	. . . . . . . . . . . . . . . . 17 ⊢ (∅ × {𝑧}) = ∅
58	56, 57	eqtrdi 2820	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
59	58	rneqd 5926	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ran ∅)
60		rn0 5914	. . . . . . . . . . . . . . 15 ⊢ ran ∅ = ∅
61	59, 60	eqtrdi 2820	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
62		0ss 4363	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ 𝐵
63	61, 62	eqsstrdi 3989	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
64	63	a1d 26	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
65		rnxp 6166	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
66	65	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
67		snssi 4753	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐵 → {𝑧} ⊆ 𝐵)
68	67	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → {𝑧} ⊆ 𝐵)
69	66, 68	eqsstrd 3979	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
70	69	ex 417	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
71	64, 70	pm2.61ine 3047	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
72		ssequn2 4150	. . . . . . . . . . 11 ⊢ (ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵 ↔ (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
73	71, 72	sylib 221	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐵 → (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
74	55, 73	sylan9eqr 2826	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
75	51, 74	eqtrid 2816	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
76		df-fo 6540	. . . . . . . 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 594	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵)
78		foeq1 6786	. . . . . . . 8 ⊢ (𝑓 = (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝐴–onto→𝐵 ↔ (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵))
79	78	spcegv 3565	. . . . . . 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 41	. . . . . 6 ⊢ (𝐴 ∈ V → ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵))
81	80	expdimp 457	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝐵) → (𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵))
82	81	exlimdv 1960	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝐵) → (∃𝑔 𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵))
83	82	ex 417	. . 3 ⊢ (𝐴 ∈ V → (𝑧 ∈ 𝐵 → (∃𝑔 𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
84	83	exlimdv 1960	. 2 ⊢ (𝐴 ∈ V → (∃𝑧 𝑧 ∈ 𝐵 → (∃𝑔 𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
85	3, 9, 11, 84	syl3c 67	1 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  {csn 4591   class class class wbr 5110   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660  Fun wfun 6528   Fn wfn 6529  ⟶wf 6530  –1-1→wf1 6531  –onto→wfo 6532   ≼ cdom 8937   ≺ csdm 8938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-en 8940  df-dom 8941  df-sdom 8942

This theorem is referenced by:  pwdom  9113  domwdom  9532  iunfictbso  10094  fodomb  10506  brdom3  10508  konigthlem  10549  1stcfb  23567  ovoliunnul  25631  sigapildsys  34493  carsgclctunlem3  34651  ovoliunnfl  38196  voliunnfl  38198  volsupnfl  38199  modelaxreplem1  45574  nnfoctb  45655  caragenunicl  47125