Theorem dffo4 5825

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 5594	. . 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 2816	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
4	3	elrn 5000	. . . . . . . . 9 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦)
5		eleq2 2296	. . . . . . . . 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 5508	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
10		fnbr 5460	. . . . . . . . . . . 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 1929	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
15		df-rex 2526	. . . . . . . 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 2615	. . . 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 5715	. . . . . . . . 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 2234	. . . . . . . 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 2644	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
28	27	ralimdv 2610	. . . 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 5824	. . 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 1398  ∃wex 1541   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521   class class class wbr 4109  ran crn 4750   Fn wfn 5347  ⟶wf 5348  –onto→wfo 5350  ‘cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360

This theorem is referenced by:  dffo5  5826