Theorem dff3 7120

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

Assertion

Ref	Expression
dff3	⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦))

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

Proof of Theorem dff3

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fssxp 6763	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
2		ffun 6739	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
3		fdm 6745	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4	3	eleq2d 2827	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
5	4	biimpar 477	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
6		funfvop 7070	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
7	2, 5, 6	syl2an2r 685	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
8		df-br 5144	. . . . . . 7 ⊢ (𝑥𝐹(𝐹‘𝑥) ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
9	7, 8	sylibr 234	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥))
10		fvex 6919	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
11		breq2 5147	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥𝐹𝑦 ↔ 𝑥𝐹(𝐹‘𝑥)))
12	10, 11	spcev 3606	. . . . . 6 ⊢ (𝑥𝐹(𝐹‘𝑥) → ∃𝑦 𝑥𝐹𝑦)
13	9, 12	syl 17	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 𝑥𝐹𝑦)
14		funmo 6581	. . . . . . 7 ⊢ (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
15	2, 14	syl 17	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ∃*𝑦 𝑥𝐹𝑦)
16	15	adantr 480	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃*𝑦 𝑥𝐹𝑦)
17		df-eu 2569	. . . . 5 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
18	13, 16, 17	sylanbrc 583	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 𝑥𝐹𝑦)
19	18	ralrimiva 3146	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)
20	1, 19	jca 511	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦))
21		xpss 5701	. . . . . . . 8 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
22		sstr 3992	. . . . . . . 8 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (V × V)) → 𝐹 ⊆ (V × V))
23	21, 22	mpan2 691	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → 𝐹 ⊆ (V × V))
24		df-rel 5692	. . . . . . 7 ⊢ (Rel 𝐹 ↔ 𝐹 ⊆ (V × V))
25	23, 24	sylibr 234	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → Rel 𝐹)
26	25	adantr 480	. . . . 5 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → Rel 𝐹)
27		df-ral 3062	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦))
28		eumo 2578	. . . . . . . . . . . 12 ⊢ (∃!𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
29	28	imim2i 16	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦) → (𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
30	29	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
31		df-br 5144	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
32		ssel 3977	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
33	31, 32	biimtrid 242	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
34		opelxp1 5727	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
35	33, 34	syl6 35	. . . . . . . . . . . . . 14 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
36	35	exlimdv 1933	. . . . . . . . . . . . 13 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∃𝑦 𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
37	36	con3d 152	. . . . . . . . . . . 12 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥 ∈ 𝐴 → ¬ ∃𝑦 𝑥𝐹𝑦))
38		nexmo 2541	. . . . . . . . . . . 12 ⊢ (¬ ∃𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
39	37, 38	syl6 35	. . . . . . . . . . 11 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
40	39	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (¬ 𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
41	30, 40	pm2.61d 179	. . . . . . . . 9 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → ∃*𝑦 𝑥𝐹𝑦)
42	41	ex 412	. . . . . . . 8 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ((𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∃*𝑦 𝑥𝐹𝑦))
43	42	alimdv 1916	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥(𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
44	27, 43	biimtrid 242	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
45	44	imp 406	. . . . 5 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
46		dffun6 6574	. . . . 5 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
47	26, 45, 46	sylanbrc 583	. . . 4 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → Fun 𝐹)
48		dmss 5913	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
49		dmxpss 6191	. . . . . . 7 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
50	48, 49	sstrdi 3996	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ 𝐴)
51		breq1 5146	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥𝐹𝑦 ↔ 𝑧𝐹𝑦))
52	51	eubidv 2586	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑧𝐹𝑦))
53	52	rspccv 3619	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧 ∈ 𝐴 → ∃!𝑦 𝑧𝐹𝑦))
54		euex 2577	. . . . . . . . 9 ⊢ (∃!𝑦 𝑧𝐹𝑦 → ∃𝑦 𝑧𝐹𝑦)
55		vex 3484	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
56	55	eldm 5911	. . . . . . . . 9 ⊢ (𝑧 ∈ dom 𝐹 ↔ ∃𝑦 𝑧𝐹𝑦)
57	54, 56	sylibr 234	. . . . . . . 8 ⊢ (∃!𝑦 𝑧𝐹𝑦 → 𝑧 ∈ dom 𝐹)
58	53, 57	syl6 35	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧 ∈ 𝐴 → 𝑧 ∈ dom 𝐹))
59	58	ssrdv 3989	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 → 𝐴 ⊆ dom 𝐹)
60	50, 59	anim12i 613	. . . . 5 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → (dom 𝐹 ⊆ 𝐴 ∧ 𝐴 ⊆ dom 𝐹))
61		eqss 3999	. . . . 5 ⊢ (dom 𝐹 = 𝐴 ↔ (dom 𝐹 ⊆ 𝐴 ∧ 𝐴 ⊆ dom 𝐹))
62	60, 61	sylibr 234	. . . 4 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → dom 𝐹 = 𝐴)
63		df-fn 6564	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
64	47, 62, 63	sylanbrc 583	. . 3 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹 Fn 𝐴)
65		rnss 5950	. . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
66		rnxpss 6192	. . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
67	65, 66	sstrdi 3996	. . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ 𝐵)
68	67	adantr 480	. . 3 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → ran 𝐹 ⊆ 𝐵)
69		df-f 6565	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
70	64, 68, 69	sylanbrc 583	. 2 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹:𝐴⟶𝐵)
71	20, 70	impbii 209	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃*wmo 2538  ∃!weu 2568  ∀wral 3061  Vcvv 3480   ⊆ wss 3951  ⟨cop 4632   class class class wbr 5143   × cxp 5683  dom cdm 5685  ran crn 5686  Rel wrel 5690  Fun wfun 6555   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569

This theorem is referenced by:  dff4  7121  seqomlem2  8491