Theorem dom2d 4404

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	|- (ph -> (x e. A -> C e. B))
dom2d.2	|- (ph -> ((x e. A /\ y e. A) -> (C = D <-> x = y)))

Assertion

Ref	Expression
dom2d	|- (ph -> (A e. R -> A ~<_ B))

Distinct variable groups:   x,y,A   x,B,y   y,C   x,D   ph,x,y

Proof of Theorem dom2d

Step	Hyp	Ref	Expression
1		f1domg 4396	. 2 |- (A e. R -> ({<.x, y>. | (x e. A /\ y = C)}:A-1-1->B -> A ~<_ B))
2		dom2d.1	. . . . . 6 |- (ph -> (x e. A -> C e. B))
3	2	r19.21aiv 1713	. . . . 5 |- (ph -> A.x e. A C e. B)
4		eqid 1475	. . . . . 6 |- {<.x, y>. | (x e. A /\ y = C)} = {<.x, y>. | (x e. A /\ y = C)}
5	4	fopab2 3823	. . . . 5 |- (A.x e. A C e. B <-> {<.x, y>. | (x e. A /\ y = C)}:A-->B)
6	3, 5	sylib 198	. . . 4 |- (ph -> {<.x, y>. | (x e. A /\ y = C)}:A-->B)
7	2	imp 350	. . . . . . . . . 10 |- ((ph /\ x e. A) -> C e. B)
8		fvopab2 3791	. . . . . . . . . . 11 |- ((x e. A /\ C e. B) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
9	8	adantll 392	. . . . . . . . . 10 |- (((ph /\ x e. A) /\ C e. B) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
10	7, 9	mpdan 704	. . . . . . . . 9 |- ((ph /\ x e. A) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
11	10	adantrr 395	. . . . . . . 8 |- ((ph /\ (x e. A /\ y e. A)) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
12		ax-17 971	. . . . . . . . . . 11 |- ((ph /\ y e. A) -> A.x(ph /\ y e. A))
13		hbopab1 2813	. . . . . . . . . . . . 13 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.x z e. {<.x, y>. | (x e. A /\ y = C)})
14		ax-17 971	. . . . . . . . . . . . 13 |- (z e. y -> A.x z e. y)
15	13, 14	hbfv 3729	. . . . . . . . . . . 12 |- (z e. ({<.x, y>. | (x e. A /\ y = C)}` y) -> A.x z e. ({<.x, y>. | (x e. A /\ y = C)}` y))
16		ax-17 971	. . . . . . . . . . . 12 |- (z e. D -> A.x z e. D)
17	15, 16	hbeq 1565	. . . . . . . . . . 11 |- (({<.x, y>. | (x e. A /\ y = C)}` y) = D -> A.x({<.x, y>. | (x e. A /\ y = C)}` y) = D)
18	12, 17	hbim 1007	. . . . . . . . . 10 |- (((ph /\ y e. A) -> ({<.x, y>. | (x e. A /\ y = C)}` y) = D) -> A.x((ph /\ y e. A) -> ({<.x, y>. | (x e. A /\ y = C)}` y) = D))
19		eleq1 1534	. . . . . . . . . . . . 13 |- (x = y -> (x e. A <-> y e. A))
20	19	anbi2d 616	. . . . . . . . . . . 12 |- (x = y -> ((ph /\ x e. A) <-> (ph /\ y e. A)))
21	20	imbi1d 613	. . . . . . . . . . 11 |- (x = y -> (((ph /\ x e. A) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C) <-> ((ph /\ y e. A) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)))
22	19	anbi1d 617	. . . . . . . . . . . . . . 15 |- (x = y -> ((x e. A /\ y e. A) <-> (y e. A /\ y e. A)))
23		anidm 432	. . . . . . . . . . . . . . 15 |- ((y e. A /\ y e. A) <-> y e. A)
24	22, 23	syl6bb 536	. . . . . . . . . . . . . 14 |- (x = y -> ((x e. A /\ y e. A) <-> y e. A))
25	24	anbi2d 616	. . . . . . . . . . . . 13 |- (x = y -> ((ph /\ (x e. A /\ y e. A)) <-> (ph /\ y e. A)))
26		fveq2 3724	. . . . . . . . . . . . . . . 16 |- (x = y -> ({<.x, y>. | (x e. A /\ y = C)}` x) = ({<.x, y>. | (x e. A /\ y = C)}` y))
27	26	adantr 389	. . . . . . . . . . . . . . 15 |- ((x = y /\ (ph /\ (x e. A /\ y e. A))) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = ({<.x, y>. | (x e. A /\ y = C)}` y))
28		dom2d.2	. . . . . . . . . . . . . . . . 17 |- (ph -> ((x e. A /\ y e. A) -> (C = D <-> x = y)))
29	28	imp 350	. . . . . . . . . . . . . . . 16 |- ((ph /\ (x e. A /\ y e. A)) -> (C = D <-> x = y))
30	29	biimparc 419	. . . . . . . . . . . . . . 15 |- ((x = y /\ (ph /\ (x e. A /\ y e. A))) -> C = D)
31	27, 30	eqeq12d 1489	. . . . . . . . . . . . . 14 |- ((x = y /\ (ph /\ (x e. A /\ y e. A))) -> (({<.x, y>. | (x e. A /\ y = C)}` x) = C <-> ({<.x, y>. | (x e. A /\ y = C)}` y) = D))
32	31	ex 373	. . . . . . . . . . . . 13 |- (x = y -> ((ph /\ (x e. A /\ y e. A)) -> (({<.x, y>. | (x e. A /\ y = C)}` x) = C <-> ({<.x, y>. | (x e. A /\ y = C)}` y) = D)))
33	25, 32	sylbird 205	. . . . . . . . . . . 12 |- (x = y -> ((ph /\ y e. A) -> (({<.x, y>. | (x e. A /\ y = C)}` x) = C <-> ({<.x, y>. | (x e. A /\ y = C)}` y) = D)))
34	33	pm5.74d 585	. . . . . . . . . . 11 |- (x = y -> (((ph /\ y e. A) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C) <-> ((ph /\ y e. A) -> ({<.x, y>. | (x e. A /\ y = C)}` y) = D)))
35	21, 34	bitrd 528	. . . . . . . . . 10 |- (x = y -> (((ph /\ x e. A) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C) <-> ((ph /\ y e. A) -> ({<.x, y>. | (x e. A /\ y = C)}` y) = D)))
36	18, 35, 10	chvar 1167	. . . . . . . . 9 |- ((ph /\ y e. A) -> ({<.x, y>. | (x e. A /\ y = C)}` y) = D)
37	36	adantrl 394	. . . . . . . 8 |- ((ph /\ (x e. A /\ y e. A)) -> ({<.x, y>. | (x e. A /\ y = C)}` y) = D)
38	11, 37	eqeq12d 1489	. . . . . . 7 |- ((ph /\ (x e. A /\ y e. A)) -> (({<.x, y>. | (x e. A /\ y = C)}` x) = ({<.x, y>. | (x e. A /\ y = C)}` y) <-> C = D))
39	29	biimpd 153	. . . . . . 7 |- ((ph /\ (x e. A /\ y e. A)) -> (C = D -> x = y))
40	38, 39	sylbid 203	. . . . . 6 |- ((ph /\ (x e. A /\ y e. A)) -> (({<.x, y>. | (x e. A /\ y = C)}` x) = ({<.x, y>. | (x e. A /\ y = C)}` y) -> x = y))
41	40	ex 373	. . . . 5 |- (ph -> ((x e. A /\ y e. A) -> (({<.x, y>. | (x e. A /\ y = C)}` x) = ({<.x, y>. | (x e. A /\ y = C)}` y) -> x = y)))
42	41	r19.21aivv 1720	. . . 4 |- (ph -> A.x e. A A.y e. A (({<.x, y>. | (x e. A /\ y = C)}` x) = ({<.x, y>. | (x e. A /\ y = C)}` y) -> x = y))
43	6, 42	jca 288	. . 3 |- (ph -> ({<.x, y>. | (x e. A /\ y = C)}:A-->B /\ A.x e. A A.y e. A (({<.x, y>. | (x e. A /\ y = C)}` x) = ({<.x, y>. | (x e. A /\ y = C)}` y) -> x = y)))
44		hbopab2 2814	. . . 4 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.y z e. {<.x, y>. | (x e. A /\ y = C)})
45	13, 44	f1fvf 3875	. . 3 |- ({<.x, y>. | (x e. A /\ y = C)}:A-1-1->B <-> ({<.x, y>. | (x e. A /\ y = C)}:A-->B /\ A.x e. A A.y e. A (({<.x, y>. | (x e. A /\ y = C)}` x) = ({<.x, y>. | (x e. A /\ y = C)}` y) -> x = y)))
46	43, 45	sylibr 200	. 2 |- (ph -> {<.x, y>. | (x e. A /\ y = C)}:A-1-1->B)
47	1, 46	syl5com 52	1