Theorem dff3 6372

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 6060	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
2		ffun 6048	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
3		fdm 6051	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4	3	eleq2d 2687	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
5	4	biimpar 502	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
6		funfvop 6329	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
7	2, 5, 6	syl2an2r 876	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
8		df-br 4654	. . . . . . 7 ⊢ (𝑥𝐹(𝐹‘𝑥) ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
9	7, 8	sylibr 224	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥))
10		fvex 6201	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
11		breq2 4657	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥𝐹𝑦 ↔ 𝑥𝐹(𝐹‘𝑥)))
12	10, 11	spcev 3300	. . . . . 6 ⊢ (𝑥𝐹(𝐹‘𝑥) → ∃𝑦 𝑥𝐹𝑦)
13	9, 12	syl 17	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 𝑥𝐹𝑦)
14		funmo 5904	. . . . . . 7 ⊢ (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
15	2, 14	syl 17	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ∃*𝑦 𝑥𝐹𝑦)
16	15	adantr 481	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃*𝑦 𝑥𝐹𝑦)
17		eu5 2496	. . . . 5 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
18	13, 16, 17	sylanbrc 698	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 𝑥𝐹𝑦)
19	18	ralrimiva 2966	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)
20	1, 19	jca 554	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦))
21		xpss 5226	. . . . . . . 8 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
22		sstr 3611	. . . . . . . 8 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (V × V)) → 𝐹 ⊆ (V × V))
23	21, 22	mpan2 707	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → 𝐹 ⊆ (V × V))
24		df-rel 5121	. . . . . . 7 ⊢ (Rel 𝐹 ↔ 𝐹 ⊆ (V × V))
25	23, 24	sylibr 224	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → Rel 𝐹)
26	25	adantr 481	. . . . 5 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → Rel 𝐹)
27		df-ral 2917	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦))
28		eumo 2499	. . . . . . . . . . . 12 ⊢ (∃!𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
29	28	imim2i 16	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦) → (𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
30	29	adantl 482	. . . . . . . . . 10 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
31		df-br 4654	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
32		ssel 3597	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
33	31, 32	syl5bi 232	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
34		opelxp1 5150	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
35	33, 34	syl6 35	. . . . . . . . . . . . . 14 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
36	35	exlimdv 1861	. . . . . . . . . . . . 13 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∃𝑦 𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
37	36	con3d 148	. . . . . . . . . . . 12 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥 ∈ 𝐴 → ¬ ∃𝑦 𝑥𝐹𝑦))
38		exmo 2495	. . . . . . . . . . . . 13 ⊢ (∃𝑦 𝑥𝐹𝑦 ∨ ∃*𝑦 𝑥𝐹𝑦)
39	38	ori 390	. . . . . . . . . . . 12 ⊢ (¬ ∃𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
40	37, 39	syl6 35	. . . . . . . . . . 11 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
41	40	adantr 481	. . . . . . . . . 10 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (¬ 𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
42	30, 41	pm2.61d 170	. . . . . . . . 9 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → ∃*𝑦 𝑥𝐹𝑦)
43	42	ex 450	. . . . . . . 8 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ((𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∃*𝑦 𝑥𝐹𝑦))
44	43	alimdv 1845	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥(𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
45	27, 44	syl5bi 232	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
46	45	imp 445	. . . . 5 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
47		dffun6 5903	. . . . 5 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
48	26, 46, 47	sylanbrc 698	. . . 4 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → Fun 𝐹)
49		dmss 5323	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
50		dmxpss 5565	. . . . . . 7 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
51	49, 50	syl6ss 3615	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ 𝐴)
52		breq1 4656	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥𝐹𝑦 ↔ 𝑧𝐹𝑦))
53	52	eubidv 2490	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑧𝐹𝑦))
54	53	rspccv 3306	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧 ∈ 𝐴 → ∃!𝑦 𝑧𝐹𝑦))
55		euex 2494	. . . . . . . . 9 ⊢ (∃!𝑦 𝑧𝐹𝑦 → ∃𝑦 𝑧𝐹𝑦)
56		vex 3203	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
57	56	eldm 5321	. . . . . . . . 9 ⊢ (𝑧 ∈ dom 𝐹 ↔ ∃𝑦 𝑧𝐹𝑦)
58	55, 57	sylibr 224	. . . . . . . 8 ⊢ (∃!𝑦 𝑧𝐹𝑦 → 𝑧 ∈ dom 𝐹)
59	54, 58	syl6 35	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧 ∈ 𝐴 → 𝑧 ∈ dom 𝐹))
60	59	ssrdv 3609	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 → 𝐴 ⊆ dom 𝐹)
61	51, 60	anim12i 590	. . . . 5 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → (dom 𝐹 ⊆ 𝐴 ∧ 𝐴 ⊆ dom 𝐹))
62		eqss 3618	. . . . 5 ⊢ (dom 𝐹 = 𝐴 ↔ (dom 𝐹 ⊆ 𝐴 ∧ 𝐴 ⊆ dom 𝐹))
63	61, 62	sylibr 224	. . . 4 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → dom 𝐹 = 𝐴)
64		df-fn 5891	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
65	48, 63, 64	sylanbrc 698	. . 3 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹 Fn 𝐴)
66		rnss 5354	. . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
67		rnxpss 5566	. . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
68	66, 67	syl6ss 3615	. . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ 𝐵)
69	68	adantr 481	. . 3 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → ran 𝐹 ⊆ 𝐵)
70		df-f 5892	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
71	65, 69, 70	sylanbrc 698	. 2 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹:𝐴⟶𝐵)
72	20, 71	impbii 199	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  ∃*wmo 2471  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ⟨cop 4183   class class class wbr 4653   × cxp 5112  dom cdm 5114  ran crn 5115  Rel wrel 5119  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  dff4  6373  seqomlem2  7546