Theorem fundmen 6772

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 4870	. . 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 5597	. . 3 |- ( ( Fun F /\ x e. dom F ) -> <. x , ( F ` x ) >. e. F )
6	5	ex 114	. 2 |- ( Fun F -> ( x e. dom F -> <. x , ( F ` x ) >. e. F ) )
7		funrel 5205	. . 3 |- ( Fun F -> Rel F )
8		elreldm 4830	. . . 4 |- ( ( Rel F /\ y e. F ) -> |^| |^| y e. dom F )
9	8	ex 114	. . 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 4611	. . . . . . . . 9 |- ( Rel F <-> F C_ ( _V X. _V ) )
12	7, 11	sylib 121	. . . . . . . 8 |- ( Fun F -> F C_ ( _V X. _V ) )
13	12	sselda 3142	. . . . . . 7 |- ( ( Fun F /\ y e. F ) -> y e. ( _V X. _V ) )
14		elvv 4666	. . . . . . 7 |- ( y e. ( _V X. _V ) <-> E. z E. w y = <. z , w >. )
15	13, 14	sylib 121	. . . . . 6 |- ( ( Fun F /\ y e. F ) -> E. z E. w y = <. z , w >. )
16		inteq 3827	. . . . . . . . . . . . . . . . 17 |- ( y = <. z , w >. -> |^| y = |^| <. z , w >. )
17	16	inteqd 3829	. . . . . . . . . . . . . . . 16 |- ( y = <. z , w >. -> |^| |^| y = |^| |^| <. z , w >. )
18		vex 2729	. . . . . . . . . . . . . . . . 17 |- z e. _V
19		vex 2729	. . . . . . . . . . . . . . . . 17 |- w e. _V
20	18, 19	op1stb 4456	. . . . . . . . . . . . . . . 16 |- |^| |^| <. z , w >. = z
21	17, 20	eqtrdi 2215	. . . . . . . . . . . . . . 15 |- ( y = <. z , w >. -> |^| |^| y = z )
22		eqeq1 2172	. . . . . . . . . . . . . . 15 |- ( x = |^| |^| y -> ( x = z <-> |^| |^| y = z ) )
23	21, 22	syl5ibr 155	. . . . . . . . . . . . . 14 |- ( x = |^| |^| y -> ( y = <. z , w >. -> x = z ) )
24		opeq1 3758	. . . . . . . . . . . . . 14 |- ( x = z -> <. x , w >. = <. z , w >. )
25	23, 24	syl6 33	. . . . . . . . . . . . 13 |- ( x = |^| |^| y -> ( y = <. z , w >. -> <. x , w >. = <. z , w >. ) )
26	25	imp 123	. . . . . . . . . . . 12 |- ( ( x = |^| |^| y /\ y = <. z , w >. ) -> <. x , w >. = <. z , w >. )
27		eqeq2 2175	. . . . . . . . . . . . . 14 |- ( <. x , w >. = <. z , w >. -> ( y = <. x , w >. <-> y = <. z , w >. ) )
28	27	biimprcd 159	. . . . . . . . . . . . 13 |- ( y = <. z , w >. -> ( <. x , w >. = <. z , w >. -> y = <. x , w >. ) )
29	28	adantl 275	. . . . . . . . . . . 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 266	. . . . . . . . . 10 |- ( ( y = <. z , w >. /\ x = |^| |^| y ) -> y = <. x , w >. )
32	31	adantl 275	. . . . . . . . 9 |- ( ( ( Fun F /\ y e. F ) /\ ( y = <. z , w >. /\ x = |^| |^| y ) ) -> y = <. x , w >. )
33	30	eleq1d 2235	. . . . . . . . . . . . . . 15 |- ( ( x = |^| |^| y /\ y = <. z , w >. ) -> ( y e. F <-> <. x , w >. e. F ) )
34	33	adantl 275	. . . . . . . . . . . . . 14 |- ( ( Fun F /\ ( x = |^| |^| y /\ y = <. z , w >. ) ) -> ( y e. F <-> <. x , w >. e. F ) )
35		funopfv 5526	. . . . . . . . . . . . . . 15 |- ( Fun F -> ( <. x , w >. e. F -> ( F ` x ) = w ) )
36	35	adantr 274	. . . . . . . . . . . . . 14 |- ( ( Fun F /\ ( x = |^| |^| y /\ y = <. z , w >. ) ) -> ( <. x , w >. e. F -> ( F ` x ) = w ) )
37	34, 36	sylbid 149	. . . . . . . . . . . . 13 |- ( ( Fun F /\ ( x = |^| |^| y /\ y = <. z , w >. ) ) -> ( y e. F -> ( F ` x ) = w ) )
38	37	exp32 363	. . . . . . . . . . . 12 |- ( Fun F -> ( x = |^| |^| y -> ( y = <. z , w >. -> ( y e. F -> ( F ` x ) = w ) ) ) )
39	38	com24 87	. . . . . . . . . . 11 |- ( Fun F -> ( y e. F -> ( y = <. z , w >. -> ( x = |^| |^| y -> ( F ` x ) = w ) ) ) )
40	39	imp43 353	. . . . . . . . . 10 |- ( ( ( Fun F /\ y e. F ) /\ ( y = <. z , w >. /\ x = |^| |^| y ) ) -> ( F ` x ) = w )
41	40	opeq2d 3765	. . . . . . . . 9 |- ( ( ( Fun F /\ y e. F ) /\ ( y = <. z , w >. /\ x = |^| |^| y ) ) -> <. x , ( F ` x ) >. = <. x , w >. )
42	32, 41	eqtr4d 2201	. . . . . . . 8 |- ( ( ( Fun F /\ y e. F ) /\ ( y = <. z , w >. /\ x = |^| |^| y ) ) -> y = <. x , ( F ` x ) >. )
43	42	exp32 363	. . . . . . 7 |- ( ( Fun F /\ y e. F ) -> ( y = <. z , w >. -> ( x = |^| |^| y -> y = <. x , ( F ` x ) >. ) ) )
44	43	exlimdvv 1885	. . . . . 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 470	. . . 4 |- ( ( Fun F /\ ( x e. dom F /\ y e. F ) ) -> ( x = |^| |^| y -> y = <. x , ( F ` x ) >. ) )
47		inteq 3827	. . . . . . . . 9 |- ( y = <. x , ( F ` x ) >. -> |^| y = |^| <. x , ( F ` x ) >. )
48	47	inteqd 3829	. . . . . . . 8 |- ( y = <. x , ( F ` x ) >. -> |^| |^| y = |^| |^| <. x , ( F ` x ) >. )
49	48	adantl 275	. . . . . . 7 |- ( ( ( Fun F /\ x e. dom F ) /\ y = <. x , ( F ` x ) >. ) -> |^| |^| y = |^| |^| <. x , ( F ` x ) >. )
50		vex 2729	. . . . . . . . 9 |- x e. _V
51		funfvex 5503	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
52		op1stbg 4457	. . . . . . . . 9 |- ( ( x e. _V /\ ( F ` x ) e. _V ) -> |^| |^| <. x , ( F ` x ) >. = x )
53	50, 51, 52	sylancr 411	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> |^| |^| <. x , ( F ` x ) >. = x )
54	53	adantr 274	. . . . . . 7 |- ( ( ( Fun F /\ x e. dom F ) /\ y = <. x , ( F ` x ) >. ) -> |^| |^| <. x , ( F ` x ) >. = x )
55	49, 54	eqtr2d 2199	. . . . . 6 |- ( ( ( Fun F /\ x e. dom F ) /\ y = <. x , ( F ` x ) >. ) -> x = |^| |^| y )
56	55	ex 114	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( y = <. x , ( F ` x ) >. -> x = |^| |^| y ) )
57	56	adantrr 471	. . . 4 |- ( ( Fun F /\ ( x e. dom F /\ y e. F ) ) -> ( y = <. x , ( F ` x ) >. -> x = |^| |^| y ) )
58	46, 57	impbid 128	. . 3 |- ( ( Fun F /\ ( x e. dom F /\ y e. F ) ) -> ( x = |^| |^| y <-> y = <. x , ( F ` x ) >. ) )
59	58	ex 114	. 2 |- ( Fun F -> ( ( x e. dom F /\ y e. F ) -> ( x = |^| |^| y <-> y = <. x , ( F ` x ) >. ) ) )
60	3, 4, 6, 10, 59	en3d 6735	1 |- ( Fun F -> dom F ~~ F )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   C_ wss 3116   <.cop 3579   |^|cint 3824   class class class wbr 3982   X. cxp 4602   dom cdm 4604   Rel wrel 4609   Fun wfun 5182   ` cfv 5188   ~~ cen 6704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-en 6707

This theorem is referenced by:  fundmeng  6773