Theorem dom2d 5463

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

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 5455	. 2 |- (A e. R -> ({<.x, y>. | (x e. A /\ y = C)}:A-1-1->B -> A ~<_ B))
2		hbopab1 3562	. . . 4 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.x z e. {<.x, y>. | (x e. A /\ y = C)})
3		hbopab2 3563	. . . 4 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.y z e. {<.x, y>. | (x e. A /\ y = C)})
4	2, 3	dff13f 4851	. . 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)))
5		dom2d.1	. . . . 5 |- (ph -> (x e. A -> C e. B))
6	5	r19.21aiv 2175	. . . 4 |- (ph -> A.x e. A C e. B)
7		eqid 1884	. . . . 5 |- {<.x, y>. | (x e. A /\ y = C)} = {<.x, y>. | (x e. A /\ y = C)}
8	7	fopab2 4796	. . . 4 |- (A.x e. A C e. B <-> {<.x, y>. | (x e. A /\ y = C)}:A-->B)
9	6, 8	sylib 215	. . 3 |- (ph -> {<.x, y>. | (x e. A /\ y = C)}:A-->B)
10	5	imp 377	. . . . . . . . 9 |- ((ph /\ x e. A) -> C e. B)
11		fvopab2 4754	. . . . . . . . . 10 |- ((x e. A /\ C e. B) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
12	11	adantll 428	. . . . . . . . 9 |- (((ph /\ x e. A) /\ C e. B) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
13	10, 12	mpdan 768	. . . . . . . 8 |- ((ph /\ x e. A) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
14	13	adantrr 431	. . . . . . 7 |- ((ph /\ (x e. A /\ y e. A)) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
15		ax-17 1317	. . . . . . . . . 10 |- ((ph /\ y e. A) -> A.x(ph /\ y e. A))
16		ax-17 1317	. . . . . . . . . . . 12 |- (z e. y -> A.x z e. y)
17	2, 16	hbfv 4686	. . . . . . . . . . 11 |- (z e. ({<.x, y>. | (x e. A /\ y = C)}` y) -> A.x z e. ({<.x, y>. | (x e. A /\ y = C)}` y))
18		ax-17 1317	. . . . . . . . . . 11 |- (z e. D -> A.x z e. D)
19	17, 18	hbeq 1995	. . . . . . . . . 10 |- (({<.x, y>. | (x e. A /\ y = C)}` y) = D -> A.x({<.x, y>. | (x e. A /\ y = C)}` y) = D)
20	15, 19	hbim 1354	. . . . . . . . 9 |- (((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))
21		eleq1 1957	. . . . . . . . . . . 12 |- (x = y -> (x e. A <-> y e. A))
22	21	anbi2d 678	. . . . . . . . . . 11 |- (x = y -> ((ph /\ x e. A) <-> (ph /\ y e. A)))
23	22	imbi1d 675	. . . . . . . . . 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)}` x) = C)))
24	21	anbi1d 679	. . . . . . . . . . . . . 14 |- (x = y -> ((x e. A /\ y e. A) <-> (y e. A /\ y e. A)))
25		anidm 478	. . . . . . . . . . . . . 14 |- ((y e. A /\ y e. A) <-> y e. A)
26	24, 25	syl6bb 595	. . . . . . . . . . . . 13 |- (x = y -> ((x e. A /\ y e. A) <-> y e. A))
27	26	anbi2d 678	. . . . . . . . . . . 12 |- (x = y -> ((ph /\ (x e. A /\ y e. A)) <-> (ph /\ y e. A)))
28		fveq2 4681	. . . . . . . . . . . . . . 15 |- (x = y -> ({<.x, y>. | (x e. A /\ y = C)}` x) = ({<.x, y>. | (x e. A /\ y = C)}` y))
29	28	adantr 425	. . . . . . . . . . . . . 14 |- ((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))
30		dom2d.2	. . . . . . . . . . . . . . . 16 |- (ph -> ((x e. A /\ y e. A) -> (C = D <-> x = y)))
31	30	imp 377	. . . . . . . . . . . . . . 15 |- ((ph /\ (x e. A /\ y e. A)) -> (C = D <-> x = y))
32	31	biimparc 463	. . . . . . . . . . . . . 14 |- ((x = y /\ (ph /\ (x e. A /\ y e. A))) -> C = D)
33	29, 32	eqeq12d 1899	. . . . . . . . . . . . 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))
34	33	ex 402	. . . . . . . . . . . 12 |- (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)))
35	27, 34	sylbird 222	. . . . . . . . . . 11 |- (x = y -> ((ph /\ y e. A) -> (({<.x, y>. | (x e. A /\ y = C)}` x) = C <-> ({<.x, y>. | (x e. A /\ y = C)}` y) = D)))
36	35	pm5.74d 645	. . . . . . . . . 10 |- (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)))
37	23, 36	bitrd 587	. . . . . . . . 9 |- (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)))
38	20, 37, 13	chvar 1530	. . . . . . . 8 |- ((ph /\ y e. A) -> ({<.x, y>. | (x e. A /\ y = C)}` y) = D)
39	38	adantrl 430	. . . . . . 7 |- ((ph /\ (x e. A /\ y e. A)) -> ({<.x, y>. | (x e. A /\ y = C)}` y) = D)
40	14, 39	eqeq12d 1899	. . . . . 6 |- ((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))
41	31	biimpd 170	. . . . . 6 |- ((ph /\ (x e. A /\ y e. A)) -> (C = D -> x = y))
42	40, 41	sylbid 220	. . . . 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))
43	42	ex 402	. . . 4 |- (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)))
44	43	r19.21aivv 2183	. . 3 |- (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))
45	4, 9, 44	sylanbrc 527	. 2 |- (ph -> {<.x, y>. | (x e. A /\ y = C)}:A-1-1->B)
46	1, 45	syl5com 63	1 |- (ph -> (A e. R -> A ~<_ B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   class class class wbr 3338  {copab 3395  -->wf 3994  -1-1->wf1 3995  ` cfv 3998   ~<_ cdom 5424

This theorem is referenced by:  dom2 5464  xpdom2 5501  mapdom2 5588  unidom 5970  infmap2lem2 8849

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-en 5427  df-dom 5428

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