Theorem dom2d 3307

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its range.

Hypotheses

Ref	Expression
dom2d.1	⊢ (φ → (x ∈ A → C ∈ B))
dom2d.2	⊢ (φ → ((x ∈ A ∧ y ∈ A) → (C = D ↔ x = y)))

Assertion

Ref	Expression
dom2d	⊢ (φ → (A ∈ R → A ≼ B))

Distinct variable group(s):   x,y,A   x,B,y   y,C   x,D   φ,x,y

Proof of Theorem dom2d

Step	Hyp	Ref	Expression
1		f1domg 3299	. . 3 ⊢ (A ∈ R → ({⟨x, y⟩∣(x ∈ A ∧ y = C)}:A–1-1→B → A ≼ B))
2		dom2d.1	. . . . . . 7 ⊢ (φ → (x ∈ A → C ∈ B))
3	2	r19.21aiv 1259	. . . . . 6 ⊢ (φ → ∀x ∈ A C ∈ B)
4		cleqid 1102	. . . . . . 7 ⊢ {⟨x, y⟩∣(x ∈ A ∧ y = C)} = {⟨x, y⟩∣(x ∈ A ∧ y = C)}
5	4	fopab2 2891	. . . . . 6 ⊢ (∀x ∈ A C ∈ B ↔ {⟨x, y⟩∣(x ∈ A ∧ y = C)}:A–→B)
6	3, 5	sylib 173	. . . . 5 ⊢ (φ → {⟨x, y⟩∣(x ∈ A ∧ y = C)}:A–→B)
7	2	imp 277	. . . . . . . . . . 11 ⊢ ((φ ∧ x ∈ A) → C ∈ B)
8		fvopab2 2878	. . . . . . . . . . . 12 ⊢ ((x ∈ A ∧ C ∈ B) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = C)
9	8	adantll 309	. . . . . . . . . . 11 ⊢ (((φ ∧ x ∈ A) ∧ C ∈ B) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = C)
10	7, 9	mpdan 527	. . . . . . . . . 10 ⊢ ((φ ∧ x ∈ A) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = C)
11	10	adantrr 312	. . . . . . . . 9 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A)) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = C)
12		ax-17 925	. . . . . . . . . . . 12 ⊢ ((φ ∧ y ∈ A) → ∀x(φ ∧ y ∈ A))
13		hbopab1 2112	. . . . . . . . . . . . . 14 ⊢ (z ∈ {⟨x, y⟩∣(x ∈ A ∧ y = C)} → ∀x z ∈ {⟨x, y⟩∣(x ∈ A ∧ y = C)})
14		ax-17 925	. . . . . . . . . . . . . 14 ⊢ (z ∈ y → ∀x z ∈ y)
15	13, 14	hbfv 2837	. . . . . . . . . . . . 13 ⊢ (z ∈ ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) → ∀x z ∈ ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y))
16		ax-17 925	. . . . . . . . . . . . 13 ⊢ (z ∈ D → ∀x z ∈ D)
17	15, 16	hbeq 1171	. . . . . . . . . . . 12 ⊢ (({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) = D → ∀x({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) = D)
18	12, 17	hbim 702	. . . . . . . . . . 11 ⊢ (((φ ∧ y ∈ A) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) = D) → ∀x((φ ∧ y ∈ A) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) = D))
19		eleq1 1149	. . . . . . . . . . . . . 14 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
20	19	anbi2d 468	. . . . . . . . . . . . 13 ⊢ (x = y → ((φ ∧ x ∈ A) ↔ (φ ∧ y ∈ A)))
21	20	imbi1d 465	. . . . . . . . . . . 12 ⊢ (x = y → (((φ ∧ x ∈ A) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = C) ↔ ((φ ∧ y ∈ A) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = C)))
22	19	anbi1d 469	. . . . . . . . . . . . . . . 16 ⊢ (x = y → ((x ∈ A ∧ y ∈ A) ↔ (y ∈ A ∧ y ∈ A)))
23		anidm 331	. . . . . . . . . . . . . . . 16 ⊢ ((y ∈ A ∧ y ∈ A) ↔ y ∈ A)
24	22, 23	syl6bb 414	. . . . . . . . . . . . . . 15 ⊢ (x = y → ((x ∈ A ∧ y ∈ A) ↔ y ∈ A))
25	24	anbi2d 468	. . . . . . . . . . . . . 14 ⊢ (x = y → ((φ ∧ (x ∈ A ∧ y ∈ A)) ↔ (φ ∧ y ∈ A)))
26		fveq2 2832	. . . . . . . . . . . . . . . . 17 ⊢ (x = y → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y))
27	26	adantr 306	. . . . . . . . . . . . . . . 16 ⊢ ((x = y ∧ (φ ∧ (x ∈ A ∧ y ∈ A))) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y))
28		dom2d.2	. . . . . . . . . . . . . . . . . 18 ⊢ (φ → ((x ∈ A ∧ y ∈ A) → (C = D ↔ x = y)))
29	28	imp 277	. . . . . . . . . . . . . . . . 17 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A)) → (C = D ↔ x = y))
30	29	biimparc 327	. . . . . . . . . . . . . . . 16 ⊢ ((x = y ∧ (φ ∧ (x ∈ A ∧ y ∈ A))) → C = D)
31	27, 30	cleq12d 1115	. . . . . . . . . . . . . . 15 ⊢ ((x = y ∧ (φ ∧ (x ∈ A ∧ y ∈ A))) → (({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = C ↔ ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) = D))
32	31	exp 291	. . . . . . . . . . . . . 14 ⊢ (x = y → ((φ ∧ (x ∈ A ∧ y ∈ A)) → (({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = C ↔ ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) = D)))
33	25, 32	sylbird 180	. . . . . . . . . . . . 13 ⊢ (x = y → ((φ ∧ y ∈ A) → (({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = C ↔ ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) = D)))
34	33	pm5.74d 444	. . . . . . . . . . . 12 ⊢ (x = y → (((φ ∧ y ∈ A) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = C) ↔ ((φ ∧ y ∈ A) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) = D)))
35	21, 34	bitrd 406	. . . . . . . . . . 11 ⊢ (x = y → (((φ ∧ x ∈ A) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = C) ↔ ((φ ∧ y ∈ A) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) = D)))
36	18, 35, 10	chv2 850	. . . . . . . . . 10 ⊢ ((φ ∧ y ∈ A) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) = D)
37	36	adantrl 311	. . . . . . . . 9 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A)) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) = D)
38	11, 37	cleq12d 1115	. . . . . . . 8 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A)) → (({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) ↔ C = D))
39	29	biimpd 135	. . . . . . . 8 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A)) → (C = D → x = y))
40	38, 39	sylbid 178	. . . . . . 7 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A)) → (({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) → x = y))
41	40	exp 291	. . . . . 6 ⊢ (φ → ((x ∈ A ∧ y ∈ A) → (({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) → x = y)))
42	41	r19.21aivv 1263	. . . . 5 ⊢ (φ → ∀x ∈ A ∀y ∈ A (({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) → x = y))
43	6, 42	jca 236	. . . 4 ⊢ (φ → ({⟨x, y⟩∣(x ∈ A ∧ y = C)}:A–→B ∧ ∀x ∈ A ∀y ∈ A (({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) → x = y)))
44		hbopab2 2113	. . . . 5 ⊢ (z ∈ {⟨x, y⟩∣(x ∈ A ∧ y = C)} → ∀y z ∈ {⟨x, y⟩∣(x ∈ A ∧ y = C)})
45	13, 44	f1fvf 2917	. . . 4 ⊢ ({⟨x, y⟩∣(x ∈ A ∧ y = C)}:A–1-1→B ↔ ({⟨x, y⟩∣(x ∈ A ∧ y = C)}:A–→B ∧ ∀x ∈ A ∀y ∈ A (({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘y) → x = y)))
46	43, 45	sylibr 175	. . 3 ⊢ (φ → {⟨x, y⟩∣(x ∈ A ∧ y = C)}:A–1-1→B)
47	1, 46	syl5 22	. 2 ⊢ (A ∈ R → (φ → A ≼ B))
48	47	com12 13	1 ⊢ (φ → (A ∈ R → A ≼ B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201   class class class wbr 2054  {copab 2055  –→wf 2418  –1-1→wf1 2419   ‘cfv 2422   ≼ cdom 3272

This theorem is referenced by:  dom2 3308  xpdom2 3345  mapdom2 3389  infmap2lem2 4952

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-en 3274  df-dom 3275

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