Theorem dffo4 5707

Description: Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Assertion

Ref	Expression
dffo4	⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem dffo4

Step	Hyp	Ref	Expression
1		dffo2 5481	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))
2		simpl 109	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴⟶𝐵)
3		vex 2763	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
4	3	elrn 4906	. . . . . . . . 9 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦)
5		eleq2 2257	. . . . . . . . 9 ⊢ (ran 𝐹 = 𝐵 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ 𝐵))
6	4, 5	bitr3id 194	. . . . . . . 8 ⊢ (ran 𝐹 = 𝐵 → (∃𝑥 𝑥𝐹𝑦 ↔ 𝑦 ∈ 𝐵))
7	6	biimpar 297	. . . . . . 7 ⊢ ((ran 𝐹 = 𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 𝑥𝐹𝑦)
8	7	adantll 476	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 𝑥𝐹𝑦)
9		ffn 5404	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
10		fnbr 5357	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴)
11	10	ex 115	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
12	9, 11	syl 14	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
13	12	ancrd 326	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥𝐹𝑦 → (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
14	13	eximdv 1891	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
15		df-rex 2478	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
16	14, 15	imbitrrdi 162	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
17	16	ad2antrr 488	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦 ∈ 𝐵) → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
18	8, 17	mpd 13	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦)
19	18	ralrimiva 2567	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦)
20	2, 19	jca 306	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
21	1, 20	sylbi 121	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
22		fnbrfvb 5598	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
23	22	biimprd 158	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 → (𝐹‘𝑥) = 𝑦))
24		eqcom 2195	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
25	23, 24	imbitrdi 161	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 → 𝑦 = (𝐹‘𝑥)))
26	9, 25	sylan 283	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 → 𝑦 = (𝐹‘𝑥)))
27	26	reximdva 2596	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
28	27	ralimdv 2562	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
29	28	imdistani 445	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
30		dffo3 5706	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
31	29, 30	sylibr 134	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦) → 𝐹:𝐴–onto→𝐵)
32	21, 31	impbii 126	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473   class class class wbr 4030  ran crn 4661   Fn wfn 5250  ⟶wf 5251  –onto→wfo 5253  ‘cfv 5255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263

This theorem is referenced by:  dffo5  5708