Theorem dffo4 7078

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 6779	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))
2		simpl 482	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴⟶𝐵)
3		vex 3454	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
4	3	elrn 5860	. . . . . . . . 9 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦)
5		eleq2 2818	. . . . . . . . 9 ⊢ (ran 𝐹 = 𝐵 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ 𝐵))
6	4, 5	bitr3id 285	. . . . . . . 8 ⊢ (ran 𝐹 = 𝐵 → (∃𝑥 𝑥𝐹𝑦 ↔ 𝑦 ∈ 𝐵))
7	6	biimpar 477	. . . . . . 7 ⊢ ((ran 𝐹 = 𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 𝑥𝐹𝑦)
8	7	adantll 714	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 𝑥𝐹𝑦)
9		ffn 6691	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
10		fnbr 6629	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴)
11	10	ex 412	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
12	9, 11	syl 17	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
13	12	ancrd 551	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥𝐹𝑦 → (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
14	13	eximdv 1917	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
15		df-rex 3055	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
16	14, 15	imbitrrdi 252	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
17	16	ad2antrr 726	. . . . . 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 3126	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦)
20	2, 19	jca 511	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
21	1, 20	sylbi 217	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
22		fnbrfvb 6914	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
23	22	biimprd 248	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 → (𝐹‘𝑥) = 𝑦))
24		eqcom 2737	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
25	23, 24	imbitrdi 251	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 → 𝑦 = (𝐹‘𝑥)))
26	9, 25	sylan 580	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 → 𝑦 = (𝐹‘𝑥)))
27	26	reximdva 3147	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
28	27	ralimdv 3148	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
29	28	imdistani 568	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
30		dffo3 7077	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
31	29, 30	sylibr 234	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦) → 𝐹:𝐴–onto→𝐵)
32	21, 31	impbii 209	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   class class class wbr 5110  ran crn 5642   Fn wfn 6509  ⟶wf 6510  –onto→wfo 6512  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fo 6520  df-fv 6522

This theorem is referenced by:  dffo5  7079  exfo  7080  brdom3  10488  phpreu  37605  poimirlem26  37647