Theorem dffo4 6846

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 6569	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))
2		simpl 486	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴⟶𝐵)
3		vex 3444	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
4	3	elrn 5786	. . . . . . . . 9 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦)
5		eleq2 2878	. . . . . . . . 9 ⊢ (ran 𝐹 = 𝐵 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ 𝐵))
6	4, 5	bitr3id 288	. . . . . . . 8 ⊢ (ran 𝐹 = 𝐵 → (∃𝑥 𝑥𝐹𝑦 ↔ 𝑦 ∈ 𝐵))
7	6	biimpar 481	. . . . . . 7 ⊢ ((ran 𝐹 = 𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 𝑥𝐹𝑦)
8	7	adantll 713	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 𝑥𝐹𝑦)
9		ffn 6487	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
10		fnbr 6430	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴)
11	10	ex 416	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
12	9, 11	syl 17	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
13	12	ancrd 555	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥𝐹𝑦 → (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
14	13	eximdv 1918	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
15		df-rex 3112	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
16	14, 15	syl6ibr 255	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
17	16	ad2antrr 725	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦 ∈ 𝐵) → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
18	8, 17	mpd 15	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦)
19	18	ralrimiva 3149	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦)
20	2, 19	jca 515	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
21	1, 20	sylbi 220	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
22		fnbrfvb 6693	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
23	22	biimprd 251	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 → (𝐹‘𝑥) = 𝑦))
24		eqcom 2805	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
25	23, 24	syl6ib 254	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 → 𝑦 = (𝐹‘𝑥)))
26	9, 25	sylan 583	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 → 𝑦 = (𝐹‘𝑥)))
27	26	reximdva 3233	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
28	27	ralimdv 3145	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
29	28	imdistani 572	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
30		dffo3 6845	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
31	29, 30	sylibr 237	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦) → 𝐹:𝐴–onto→𝐵)
32	21, 31	impbii 212	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   class class class wbr 5030  ran crn 5520   Fn wfn 6319  ⟶wf 6320  –onto→wfo 6322  ‘cfv 6324

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332

This theorem is referenced by:  dffo5  6847  exfo  6848  brdom3  9939  phpreu  35041  poimirlem26  35083