Theorem fundmen 6273

Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Hypothesis

Ref	Expression
fundmen.1	|- F e. _V

Assertion

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

Proof of Theorem fundmen

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fundmen.1	. . . 4 |- F e. _V
2	1	dmex 4585	. . 3 |- dom F e. _V
3	2	a1i 9	. 2 |- ( Fun F -> dom F e. _V )
4	1	a1i 9	. 2 |- ( Fun F -> F e. _V )
5		funfvop 5266	. . 3 |- ( ( Fun F /\ x e. dom F ) -> <. x , ( F ` x ) >. e. F )
6	5	ex 108	. 2 |- ( Fun F -> ( x e. dom F -> <. x , ( F ` x ) >. e. F ) )
7		funrel 4906	. . 3 |- ( Fun F -> Rel F )
8		elreldm 4547	. . . 4 |- ( ( Rel F /\ y e. F ) -> |^| |^| y e. dom F )
9	8	ex 108	. . 3 |- ( Rel F -> ( y e. F -> |^| |^| y e. dom F ) )
10	7, 9	syl 14	. 2 |- ( Fun F -> ( y e. F -> |^| |^| y e. dom F ) )
11		df-rel 4339	. . . . . . . . 9 |- ( Rel F <-> F C_ ( _V X. _V ) )
12	7, 11	sylib 127	. . . . . . . 8 |- ( Fun F -> F C_ ( _V X. _V ) )
13	12	sselda 2942	. . . . . . 7 |- ( ( Fun F /\ y e. F ) -> y e. ( _V X. _V ) )
14		elvv 4389	. . . . . . 7 |- ( y e. ( _V X. _V ) <-> E. z E. w y = <. z , w >. )
15	13, 14	sylib 127	. . . . . 6 |- ( ( Fun F /\ y e. F ) -> E. z E. w y = <. z , w >. )
16		inteq 3615	. . . . . . . . . . . . . . . . 17 |- ( y = <. z , w >. -> |^| y = |^| <. z , w >. )
17	16	inteqd 3617	. . . . . . . . . . . . . . . 16 |- ( y = <. z , w >. -> |^| |^| y = |^| |^| <. z , w >. )
18		vex 2557	. . . . . . . . . . . . . . . . 17 |- z e. _V
19		vex 2557	. . . . . . . . . . . . . . . . 17 |- w e. _V
20	18, 19	op1stb 4205	. . . . . . . . . . . . . . . 16 |- |^| |^| <. z , w >. = z
21	17, 20	syl6eq 2088	. . . . . . . . . . . . . . 15 |- ( y = <. z , w >. -> |^| |^| y = z )
22		eqeq1 2046	. . . . . . . . . . . . . . 15 |- ( x = |^| |^| y -> ( x = z <-> |^| |^| y = z ) )
23	21, 22	syl5ibr 145	. . . . . . . . . . . . . 14 |- ( x = |^| |^| y -> ( y = <. z , w >. -> x = z ) )
24		opeq1 3546	. . . . . . . . . . . . . 14 |- ( x = z -> <. x , w >. = <. z , w >. )
25	23, 24	syl6 29	. . . . . . . . . . . . 13 |- ( x = |^| |^| y -> ( y = <. z , w >. -> <. x , w >. = <. z , w >. ) )
26	25	imp 115	. . . . . . . . . . . 12 |- ( ( x = |^| |^| y /\ y = <. z , w >. ) -> <. x , w >. = <. z , w >. )
27		eqeq2 2049	. . . . . . . . . . . . . 14 |- ( <. x , w >. = <. z , w >. -> ( y = <. x , w >. <-> y = <. z , w >. ) )
28	27	biimprcd 149	. . . . . . . . . . . . 13 |- ( y = <. z , w >. -> ( <. x , w >. = <. z , w >. -> y = <. x , w >. ) )
29	28	adantl 262	. . . . . . . . . . . 12 |- ( ( x = |^| |^| y /\ y = <. z , w >. ) -> ( <. x , w >. = <. z , w >. -> y = <. x , w >. ) )
30	26, 29	mpd 13	. . . . . . . . . . 11 |- ( ( x = |^| |^| y /\ y = <. z , w >. ) -> y = <. x , w >. )
31	30	ancoms 255	. . . . . . . . . 10 |- ( ( y = <. z , w >. /\ x = |^| |^| y ) -> y = <. x , w >. )
32	31	adantl 262	. . . . . . . . 9 |- ( ( ( Fun F /\ y e. F ) /\ ( y = <. z , w >. /\ x = |^| |^| y ) ) -> y = <. x , w >. )
33	30	eleq1d 2106	. . . . . . . . . . . . . . 15 |- ( ( x = |^| |^| y /\ y = <. z , w >. ) -> ( y e. F <-> <. x , w >. e. F ) )
34	33	adantl 262	. . . . . . . . . . . . . 14 |- ( ( Fun F /\ ( x = |^| |^| y /\ y = <. z , w >. ) ) -> ( y e. F <-> <. x , w >. e. F ) )
35		funopfv 5200	. . . . . . . . . . . . . . 15 |- ( Fun F -> ( <. x , w >. e. F -> ( F ` x ) = w ) )
36	35	adantr 261	. . . . . . . . . . . . . 14 |- ( ( Fun F /\ ( x = |^| |^| y /\ y = <. z , w >. ) ) -> ( <. x , w >. e. F -> ( F ` x ) = w ) )
37	34, 36	sylbid 139	. . . . . . . . . . . . 13 |- ( ( Fun F /\ ( x = |^| |^| y /\ y = <. z , w >. ) ) -> ( y e. F -> ( F ` x ) = w ) )
38	37	exp32 347	. . . . . . . . . . . 12 |- ( Fun F -> ( x = |^| |^| y -> ( y = <. z , w >. -> ( y e. F -> ( F ` x ) = w ) ) ) )
39	38	com24 81	. . . . . . . . . . 11 |- ( Fun F -> ( y e. F -> ( y = <. z , w >. -> ( x = |^| |^| y -> ( F ` x ) = w ) ) ) )
40	39	imp43 337	. . . . . . . . . 10 |- ( ( ( Fun F /\ y e. F ) /\ ( y = <. z , w >. /\ x = |^| |^| y ) ) -> ( F ` x ) = w )
41	40	opeq2d 3553	. . . . . . . . 9 |- ( ( ( Fun F /\ y e. F ) /\ ( y = <. z , w >. /\ x = |^| |^| y ) ) -> <. x , ( F ` x ) >. = <. x , w >. )
42	32, 41	eqtr4d 2075	. . . . . . . 8 |- ( ( ( Fun F /\ y e. F ) /\ ( y = <. z , w >. /\ x = |^| |^| y ) ) -> y = <. x , ( F ` x ) >. )
43	42	exp32 347	. . . . . . 7 |- ( ( Fun F /\ y e. F ) -> ( y = <. z , w >. -> ( x = |^| |^| y -> y = <. x , ( F ` x ) >. ) ) )
44	43	exlimdvv 1777	. . . . . 6 |- ( ( Fun F /\ y e. F ) -> ( E. z E. w y = <. z , w >. -> ( x = |^| |^| y -> y = <. x , ( F ` x ) >. ) ) )
45	15, 44	mpd 13	. . . . 5 |- ( ( Fun F /\ y e. F ) -> ( x = |^| |^| y -> y = <. x , ( F ` x ) >. ) )
46	45	adantrl 447	. . . 4 |- ( ( Fun F /\ ( x e. dom F /\ y e. F ) ) -> ( x = |^| |^| y -> y = <. x , ( F ` x ) >. ) )
47		inteq 3615	. . . . . . . . 9 |- ( y = <. x , ( F ` x ) >. -> |^| y = |^| <. x , ( F ` x ) >. )
48	47	inteqd 3617	. . . . . . . 8 |- ( y = <. x , ( F ` x ) >. -> |^| |^| y = |^| |^| <. x , ( F ` x ) >. )
49	48	adantl 262	. . . . . . 7 |- ( ( ( Fun F /\ x e. dom F ) /\ y = <. x , ( F ` x ) >. ) -> |^| |^| y = |^| |^| <. x , ( F ` x ) >. )
50		vex 2557	. . . . . . . . 9 |- x e. _V
51		funfvex 5179	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
52		op1stbg 4206	. . . . . . . . 9 |- ( ( x e. _V /\ ( F ` x ) e. _V ) -> |^| |^| <. x , ( F ` x ) >. = x )
53	50, 51, 52	sylancr 393	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> |^| |^| <. x , ( F ` x ) >. = x )
54	53	adantr 261	. . . . . . 7 |- ( ( ( Fun F /\ x e. dom F ) /\ y = <. x , ( F ` x ) >. ) -> |^| |^| <. x , ( F ` x ) >. = x )
55	49, 54	eqtr2d 2073	. . . . . 6 |- ( ( ( Fun F /\ x e. dom F ) /\ y = <. x , ( F ` x ) >. ) -> x = |^| |^| y )
56	55	ex 108	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( y = <. x , ( F ` x ) >. -> x = |^| |^| y ) )
57	56	adantrr 448	. . . 4 |- ( ( Fun F /\ ( x e. dom F /\ y e. F ) ) -> ( y = <. x , ( F ` x ) >. -> x = |^| |^| y ) )
58	46, 57	impbid 120	. . 3 |- ( ( Fun F /\ ( x e. dom F /\ y e. F ) ) -> ( x = |^| |^| y <-> y = <. x , ( F ` x ) >. ) )
59	58	ex 108	. 2 |- ( Fun F -> ( ( x e. dom F /\ y e. F ) -> ( x = |^| |^| y <-> y = <. x , ( F ` x ) >. ) ) )
60	3, 4, 6, 10, 59	en3d 6236	1 |- ( Fun F -> dom F ~~ F )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2554   C_ wss 2914   <.cop 3375   |^|cint 3612   class class class wbr 3761   X. cxp 4330   dom cdm 4332   Rel wrel 4337   Fun wfun 4883   ` cfv 4889   ~~ cen 6206

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941  ax-un 4166

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-br 3762  df-opab 3816  df-mpt 3817  df-id 4027  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-iota 4854  df-fun 4891  df-fn 4892  df-f 4893  df-f1 4894  df-fo 4895  df-f1o 4896  df-fv 4897  df-en 6209

This theorem is referenced by:  fundmeng  6274