Theorem dffo4 6961

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 6676	. . 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 3426	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
4	3	elrn 5791	. . . . . . . . 9 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦)
5		eleq2 2827	. . . . . . . . 9 ⊢ (ran 𝐹 = 𝐵 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ 𝐵))
6	4, 5	bitr3id 284	. . . . . . . 8 ⊢ (ran 𝐹 = 𝐵 → (∃𝑥 𝑥𝐹𝑦 ↔ 𝑦 ∈ 𝐵))
7	6	biimpar 477	. . . . . . 7 ⊢ ((ran 𝐹 = 𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 𝑥𝐹𝑦)
8	7	adantll 710	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 𝑥𝐹𝑦)
9		ffn 6584	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
10		fnbr 6525	. . . . . . . . . . . 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 1921	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
15		df-rex 3069	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
16	14, 15	syl6ibr 251	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
17	16	ad2antrr 722	. . . . . 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 3107	. . . 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 216	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
22		fnbrfvb 6804	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
23	22	biimprd 247	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 → (𝐹‘𝑥) = 𝑦))
24		eqcom 2745	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
25	23, 24	syl6ib 250	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 → 𝑦 = (𝐹‘𝑥)))
26	9, 25	sylan 579	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 → 𝑦 = (𝐹‘𝑥)))
27	26	reximdva 3202	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
28	27	ralimdv 3103	. . . 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 6960	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
31	29, 30	sylibr 233	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦) → 𝐹:𝐴–onto→𝐵)
32	21, 31	impbii 208	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   class class class wbr 5070  ran crn 5581   Fn wfn 6413  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426

This theorem is referenced by:  dffo5  6962  exfo  6963  brdom3  10215  phpreu  35688  poimirlem26  35730