Theorem dff3 7100

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 6737	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
2		ffun 6712	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
3		fdm 6719	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4	3	eleq2d 2846	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
5	4	biimpar 483	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
6		funfvop 7049	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
7	2, 5, 6	syl2an2r 698	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
8		df-br 5103	. . . . . . 7 ⊢ (𝑥𝐹(𝐹‘𝑥) ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
9	7, 8	sylibr 237	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥))
10		fvex 6898	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
11		breq2 5106	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥𝐹𝑦 ↔ 𝑥𝐹(𝐹‘𝑥)))
12	10, 11	spcev 3560	. . . . . 6 ⊢ (𝑥𝐹(𝐹‘𝑥) → ∃𝑦 𝑥𝐹𝑦)
13	9, 12	syl 18	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 𝑥𝐹𝑦)
14		funmo 6554	. . . . . . 7 ⊢ (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
15	2, 14	syl 18	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ∃*𝑦 𝑥𝐹𝑦)
16	15	adantr 486	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃*𝑦 𝑥𝐹𝑦)
17		df-eu 2594	. . . . 5 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
18	13, 16, 17	sylanbrc 595	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 𝑥𝐹𝑦)
19	18	ralrimiva 3154	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)
20	1, 19	jca 521	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦))
21		xpss 5666	. . . . . . . 8 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
22		sstr 3916	. . . . . . . 8 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (V × V)) → 𝐹 ⊆ (V × V))
23	21, 22	mpan2 704	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → 𝐹 ⊆ (V × V))
24		df-rel 5657	. . . . . . 7 ⊢ (Rel 𝐹 ↔ 𝐹 ⊆ (V × V))
25	23, 24	sylibr 237	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → Rel 𝐹)
26	25	adantr 486	. . . . 5 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → Rel 𝐹)
27		df-ral 3077	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦))
28		eumo 2603	. . . . . . . . . . . 12 ⊢ (∃!𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
29	28	imim2i 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦) → (𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
30	29	adantl 487	. . . . . . . . . 10 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
31		df-br 5103	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
32		ssel 3902	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
33	31, 32	biimtrid 245	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
34		opelxp1 5692	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
35	33, 34	syl6 36	. . . . . . . . . . . . . 14 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
36	35	exlimdv 1965	. . . . . . . . . . . . 13 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∃𝑦 𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
37	36	con3d 153	. . . . . . . . . . . 12 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥 ∈ 𝐴 → ¬ ∃𝑦 𝑥𝐹𝑦))
38		nexmo 2566	. . . . . . . . . . . 12 ⊢ (¬ ∃𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
39	37, 38	syl6 36	. . . . . . . . . . 11 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
40	39	adantr 486	. . . . . . . . . 10 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (¬ 𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
41	30, 40	pm2.61d 181	. . . . . . . . 9 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → ∃*𝑦 𝑥𝐹𝑦)
42	41	ex 418	. . . . . . . 8 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ((𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∃*𝑦 𝑥𝐹𝑦))
43	42	alimdv 1948	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥(𝑥 ∈ 𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
44	27, 43	biimtrid 245	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
45	44	imp 412	. . . . 5 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
46		dffun6 6549	. . . . 5 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
47	26, 45, 46	sylanbrc 595	. . . 4 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → Fun 𝐹)
48		dmss 5883	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
49		dmxpss 6162	. . . . . . 7 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
50	48, 49	sstrdi 3920	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ 𝐴)
51		breq1 5105	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥𝐹𝑦 ↔ 𝑧𝐹𝑦))
52	51	eubidv 2611	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑧𝐹𝑦))
53	52	rspccv 3573	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧 ∈ 𝐴 → ∃!𝑦 𝑧𝐹𝑦))
54		euex 2602	. . . . . . . . 9 ⊢ (∃!𝑦 𝑧𝐹𝑦 → ∃𝑦 𝑧𝐹𝑦)
55		vex 3454	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
56	55	eldm 5881	. . . . . . . . 9 ⊢ (𝑧 ∈ dom 𝐹 ↔ ∃𝑦 𝑧𝐹𝑦)
57	54, 56	sylibr 237	. . . . . . . 8 ⊢ (∃!𝑦 𝑧𝐹𝑦 → 𝑧 ∈ dom 𝐹)
58	53, 57	syl6 36	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧 ∈ 𝐴 → 𝑧 ∈ dom 𝐹))
59	58	ssrdv 3914	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 → 𝐴 ⊆ dom 𝐹)
60	50, 59	anim12i 625	. . . . 5 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → (dom 𝐹 ⊆ 𝐴 ∧ 𝐴 ⊆ dom 𝐹))
61		eqss 3923	. . . . 5 ⊢ (dom 𝐹 = 𝐴 ↔ (dom 𝐹 ⊆ 𝐴 ∧ 𝐴 ⊆ dom 𝐹))
62	60, 61	sylibr 237	. . . 4 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → dom 𝐹 = 𝐴)
63		df-fn 6541	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
64	47, 62, 63	sylanbrc 595	. . 3 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹 Fn 𝐴)
65		rnss 5920	. . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
66		rnxpss 6163	. . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
67	65, 66	sstrdi 3920	. . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ 𝐵)
68	67	adantr 486	. . 3 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → ran 𝐹 ⊆ 𝐵)
69		df-f 6542	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
70	64, 68, 69	sylanbrc 595	. 2 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹:𝐴⟶𝐵)
71	20, 70	impbii 212	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃*wmo 2562  ∃!weu 2593  ∀wral 3076  Vcvv 3450   ⊆ wss 3895  ⟨cop 4834   class class class wbr 5102   × cxp 5648  dom cdm 5650  ran crn 5651  Rel wrel 5655  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546

This theorem is used by:  dff4  7101  seqomlem2  8464