Theorem fundmen 5487

Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.

Hypothesis

Ref	Expression
fundmen.1	|- F e. _V

Assertion

Ref	Expression
fundmen	|- (Fun F -> dom F ~~ F)

Proof of Theorem fundmen

Step	Hyp	Ref	Expression
1		fundmen.1	. . . 4 |- F e. _V
2	1	dmex 4208	. . 3 |- dom F e. _V
3	2	a1i 8	. 2 |- (Fun F -> dom F e. _V)
4		funfvop 4776	. . 3 |- ((Fun F /\ x e. dom F) -> <.x, (F` x)>. e. F)
5	4	ex 402	. 2 |- (Fun F -> (x e. dom F -> <.x, (F` x)>. e. F))
6		funrel 4438	. . 3 |- (Fun F -> Rel F)
7		elreldm 4185	. . . 4 |- ((Rel F /\ y e. F) -> |^||^|y e. dom F)
8	7	ex 402	. . 3 |- (Rel F -> (y e. F -> |^||^|y e. dom F))
9	6, 8	syl 12	. 2 |- (Fun F -> (y e. F -> |^||^|y e. dom F))
10		ssel2 2616	. . . . . . . 8 |- ((F C_ (_V X. _V) /\ y e. F) -> y e. (_V X. _V))
11		df-rel 4001	. . . . . . . . 9 |- (Rel F <-> F C_ (_V X. _V))
12	6, 11	sylib 215	. . . . . . . 8 |- (Fun F -> F C_ (_V X. _V))
13	10, 12	sylan 497	. . . . . . 7 |- ((Fun F /\ y e. F) -> y e. (_V X. _V))
14		elvv 4053	. . . . . . 7 |- (y e. (_V X. _V) <-> E.zE.w y = <.z, w>.)
15	13, 14	sylib 215	. . . . . 6 |- ((Fun F /\ y e. F) -> E.zE.w y = <.z, w>.)
16		eqeq1 1890	. . . . . . . . . . . . . . 15 |- (x = |^||^|y -> (x = z <-> |^||^|y = z))
17		inteq 3217	. . . . . . . . . . . . . . . . 17 |- (y = <.z, w>. -> |^|y = |^|<.z, w>.)
18	17	inteqd 3219	. . . . . . . . . . . . . . . 16 |- (y = <.z, w>. -> |^||^|y = |^||^|<.z, w>.)
19		visset 2295	. . . . . . . . . . . . . . . . 17 |- z e. _V
20	19	op1stb 3857	. . . . . . . . . . . . . . . 16 |- |^||^|<.z, w>. = z
21	18, 20	syl6eq 1944	. . . . . . . . . . . . . . 15 |- (y = <.z, w>. -> |^||^|y = z)
22	16, 21	syl5bir 227	. . . . . . . . . . . . . 14 |- (x = |^||^|y -> (y = <.z, w>. -> x = z))
23		opeq1 3158	. . . . . . . . . . . . . 14 |- (x = z -> <.x, w>. = <.z, w>.)
24	22, 23	syl6 25	. . . . . . . . . . . . 13 |- (x = |^||^|y -> (y = <.z, w>. -> <.x, w>. = <.z, w>.))
25	24	imp 377	. . . . . . . . . . . 12 |- ((x = |^||^|y /\ y = <.z, w>.) -> <.x, w>. = <.z, w>.)
26		eqeq2 1893	. . . . . . . . . . . . . 14 |- (<.x, w>. = <.z, w>. -> (y = <.x, w>. <-> y = <.z, w>.))
27	26	biimprcd 173	. . . . . . . . . . . . 13 |- (y = <.z, w>. -> (<.x, w>. = <.z, w>. -> y = <.x, w>.))
28	27	adantl 424	. . . . . . . . . . . 12 |- ((x = |^||^|y /\ y = <.z, w>.) -> (<.x, w>. = <.z, w>. -> y = <.x, w>.))
29	25, 28	mpd 29	. . . . . . . . . . 11 |- ((x = |^||^|y /\ y = <.z, w>.) -> y = <.x, w>.)
30	29	ancoms 484	. . . . . . . . . 10 |- ((y = <.z, w>. /\ x = |^||^|y) -> y = <.x, w>.)
31	30	adantl 424	. . . . . . . . 9 |- (((Fun F /\ y e. F) /\ (y = <.z, w>. /\ x = |^||^|y)) -> y = <.x, w>.)
32	29	eleq1d 1963	. . . . . . . . . . . . . . 15 |- ((x = |^||^|y /\ y = <.z, w>.) -> (y e. F <-> <.x, w>. e. F))
33	32	adantl 424	. . . . . . . . . . . . . 14 |- ((Fun F /\ (x = |^||^|y /\ y = <.z, w>.)) -> (y e. F <-> <.x, w>. e. F))
34		visset 2295	. . . . . . . . . . . . . . . 16 |- w e. _V
35	34	funopfv 4710	. . . . . . . . . . . . . . 15 |- (Fun F -> (<.x, w>. e. F -> (F` x) = w))
36	35	adantr 425	. . . . . . . . . . . . . 14 |- ((Fun F /\ (x = |^||^|y /\ y = <.z, w>.)) -> (<.x, w>. e. F -> (F` x) = w))
37	33, 36	sylbid 220	. . . . . . . . . . . . 13 |- ((Fun F /\ (x = |^||^|y /\ y = <.z, w>.)) -> (y e. F -> (F` x) = w))
38	37	exp32 408	. . . . . . . . . . . 12 |- (Fun F -> (x = |^||^|y -> (y = <.z, w>. -> (y e. F -> (F` x) = w))))
39	38	com24 41	. . . . . . . . . . 11 |- (Fun F -> (y e. F -> (y = <.z, w>. -> (x = |^||^|y -> (F` x) = w))))
40	39	imp43 397	. . . . . . . . . 10 |- (((Fun F /\ y e. F) /\ (y = <.z, w>. /\ x = |^||^|y)) -> (F` x) = w)
41	40	opeq2d 3165	. . . . . . . . 9 |- (((Fun F /\ y e. F) /\ (y = <.z, w>. /\ x = |^||^|y)) -> <.x, (F` x)>. = <.x, w>.)
42	31, 41	eqtr4d 1928	. . . . . . . 8 |- (((Fun F /\ y e. F) /\ (y = <.z, w>. /\ x = |^||^|y)) -> y = <.x, (F` x)>.)
43	42	exp32 408	. . . . . . 7 |- ((Fun F /\ y e. F) -> (y = <.z, w>. -> (x = |^||^|y -> y = <.x, (F` x)>.)))
44	43	19.23advv 1676	. . . . . 6 |- ((Fun F /\ y e. F) -> (E.zE.w y = <.z, w>. -> (x = |^||^|y -> y = <.x, (F` x)>.)))
45	15, 44	mpd 29	. . . . 5 |- ((Fun F /\ y e. F) -> (x = |^||^|y -> y = <.x, (F` x)>.))
46	45	adantrl 430	. . . 4 |- ((Fun F /\ (x e. dom F /\ y e. F)) -> (x = |^||^|y -> y = <.x, (F` x)>.))
47		inteq 3217	. . . . . 6 |- (y = <.x, (F` x)>. -> |^|y = |^|<.x, (F` x)>.)
48	47	inteqd 3219	. . . . 5 |- (y = <.x, (F` x)>. -> |^||^|y = |^||^|<.x, (F` x)>.)
49		visset 2295	. . . . . 6 |- x e. _V
50	49	op1stb 3857	. . . . 5 |- |^||^|<.x, (F` x)>. = x
51	48, 50	syl6req 1945	. . . 4 |- (y = <.x, (F` x)>. -> x = |^||^|y)
52	46, 51	impbid1 575	. . 3 |- ((Fun F /\ (x e. dom F /\ y e. F)) -> (x = |^||^|y <-> y = <.x, (F` x)>.))
53	52	ex 402	. 2 |- (Fun F -> ((x e. dom F /\ y e. F) -> (x = |^||^|y <-> y = <.x, (F` x)>.)))
54	3, 5, 9, 53	en3d 5460	1 |- (Fun F -> dom F ~~ F)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292   C_ wss 2593  <.cop 3046  |^|cint 3214   class class class wbr 3338   X. cxp 3984  dom cdm 3986  Rel wrel 3991  Fun wfun 3992  ` cfv 3998   ~~ cen 5423

This theorem is referenced by:  infmap2 8850  fndmeng 13598  fnctartar 15284  fnctartar2 15285

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-int 3215  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

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