Theorem fundmen 7024

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 5005	. . 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 5768	. . 3 |- ( ( Fun F /\ x e. dom F ) -> <. x , ( F ` x ) >. e. F )
6	5	ex 115	. 2 |- ( Fun F -> ( x e. dom F -> <. x , ( F ` x ) >. e. F ) )
7		funrel 5350	. . 3 |- ( Fun F -> Rel F )
8		elreldm 4964	. . . 4 |- ( ( Rel F /\ y e. F ) -> |^| |^| y e. dom F )
9	8	ex 115	. . 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 4738	. . . . . . . . 9 |- ( Rel F <-> F C_ ( _V X. _V ) )
12	7, 11	sylib 122	. . . . . . . 8 |- ( Fun F -> F C_ ( _V X. _V ) )
13	12	sselda 3228	. . . . . . 7 |- ( ( Fun F /\ y e. F ) -> y e. ( _V X. _V ) )
14		elvv 4794	. . . . . . 7 |- ( y e. ( _V X. _V ) <-> E. z E. w y = <. z , w >. )
15	13, 14	sylib 122	. . . . . 6 |- ( ( Fun F /\ y e. F ) -> E. z E. w y = <. z , w >. )
16		inteq 3936	. . . . . . . . . . . . . . . . 17 |- ( y = <. z , w >. -> |^| y = |^| <. z , w >. )
17	16	inteqd 3938	. . . . . . . . . . . . . . . 16 |- ( y = <. z , w >. -> |^| |^| y = |^| |^| <. z , w >. )
18		vex 2806	. . . . . . . . . . . . . . . . 17 |- z e. _V
19		vex 2806	. . . . . . . . . . . . . . . . 17 |- w e. _V
20	18, 19	op1stb 4581	. . . . . . . . . . . . . . . 16 |- |^| |^| <. z , w >. = z
21	17, 20	eqtrdi 2280	. . . . . . . . . . . . . . 15 |- ( y = <. z , w >. -> |^| |^| y = z )
22		eqeq1 2238	. . . . . . . . . . . . . . 15 |- ( x = |^| |^| y -> ( x = z <-> |^| |^| y = z ) )
23	21, 22	imbitrrid 156	. . . . . . . . . . . . . 14 |- ( x = |^| |^| y -> ( y = <. z , w >. -> x = z ) )
24		opeq1 3867	. . . . . . . . . . . . . 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 124	. . . . . . . . . . . 12 |- ( ( x = |^| |^| y /\ y = <. z , w >. ) -> <. x , w >. = <. z , w >. )
27		eqeq2 2241	. . . . . . . . . . . . . 14 |- ( <. x , w >. = <. z , w >. -> ( y = <. x , w >. <-> y = <. z , w >. ) )
28	27	biimprcd 160	. . . . . . . . . . . . 13 |- ( y = <. z , w >. -> ( <. x , w >. = <. z , w >. -> y = <. x , w >. ) )
29	28	adantl 277	. . . . . . . . . . . 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 268	. . . . . . . . . 10 |- ( ( y = <. z , w >. /\ x = |^| |^| y ) -> y = <. x , w >. )
32	31	adantl 277	. . . . . . . . 9 |- ( ( ( Fun F /\ y e. F ) /\ ( y = <. z , w >. /\ x = |^| |^| y ) ) -> y = <. x , w >. )
33	30	eleq1d 2300	. . . . . . . . . . . . . . 15 |- ( ( x = |^| |^| y /\ y = <. z , w >. ) -> ( y e. F <-> <. x , w >. e. F ) )
34	33	adantl 277	. . . . . . . . . . . . . 14 |- ( ( Fun F /\ ( x = |^| |^| y /\ y = <. z , w >. ) ) -> ( y e. F <-> <. x , w >. e. F ) )
35		funopfv 5692	. . . . . . . . . . . . . . 15 |- ( Fun F -> ( <. x , w >. e. F -> ( F ` x ) = w ) )
36	35	adantr 276	. . . . . . . . . . . . . 14 |- ( ( Fun F /\ ( x = |^| |^| y /\ y = <. z , w >. ) ) -> ( <. x , w >. e. F -> ( F ` x ) = w ) )
37	34, 36	sylbid 150	. . . . . . . . . . . . 13 |- ( ( Fun F /\ ( x = |^| |^| y /\ y = <. z , w >. ) ) -> ( y e. F -> ( F ` x ) = w ) )
38	37	exp32 365	. . . . . . . . . . . 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 355	. . . . . . . . . 10 |- ( ( ( Fun F /\ y e. F ) /\ ( y = <. z , w >. /\ x = |^| |^| y ) ) -> ( F ` x ) = w )
41	40	opeq2d 3874	. . . . . . . . 9 |- ( ( ( Fun F /\ y e. F ) /\ ( y = <. z , w >. /\ x = |^| |^| y ) ) -> <. x , ( F ` x ) >. = <. x , w >. )
42	32, 41	eqtr4d 2267	. . . . . . . 8 |- ( ( ( Fun F /\ y e. F ) /\ ( y = <. z , w >. /\ x = |^| |^| y ) ) -> y = <. x , ( F ` x ) >. )
43	42	exp32 365	. . . . . . 7 |- ( ( Fun F /\ y e. F ) -> ( y = <. z , w >. -> ( x = |^| |^| y -> y = <. x , ( F ` x ) >. ) ) )
44	43	exlimdvv 1946	. . . . . 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 478	. . . 4 |- ( ( Fun F /\ ( x e. dom F /\ y e. F ) ) -> ( x = |^| |^| y -> y = <. x , ( F ` x ) >. ) )
47		inteq 3936	. . . . . . . . 9 |- ( y = <. x , ( F ` x ) >. -> |^| y = |^| <. x , ( F ` x ) >. )
48	47	inteqd 3938	. . . . . . . 8 |- ( y = <. x , ( F ` x ) >. -> |^| |^| y = |^| |^| <. x , ( F ` x ) >. )
49	48	adantl 277	. . . . . . 7 |- ( ( ( Fun F /\ x e. dom F ) /\ y = <. x , ( F ` x ) >. ) -> |^| |^| y = |^| |^| <. x , ( F ` x ) >. )
50		vex 2806	. . . . . . . . 9 |- x e. _V
51		funfvex 5665	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
52		op1stbg 4582	. . . . . . . . 9 |- ( ( x e. _V /\ ( F ` x ) e. _V ) -> |^| |^| <. x , ( F ` x ) >. = x )
53	50, 51, 52	sylancr 414	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> |^| |^| <. x , ( F ` x ) >. = x )
54	53	adantr 276	. . . . . . 7 |- ( ( ( Fun F /\ x e. dom F ) /\ y = <. x , ( F ` x ) >. ) -> |^| |^| <. x , ( F ` x ) >. = x )
55	49, 54	eqtr2d 2265	. . . . . 6 |- ( ( ( Fun F /\ x e. dom F ) /\ y = <. x , ( F ` x ) >. ) -> x = |^| |^| y )
56	55	ex 115	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( y = <. x , ( F ` x ) >. -> x = |^| |^| y ) )
57	56	adantrr 479	. . . 4 |- ( ( Fun F /\ ( x e. dom F /\ y e. F ) ) -> ( y = <. x , ( F ` x ) >. -> x = |^| |^| y ) )
58	46, 57	impbid 129	. . 3 |- ( ( Fun F /\ ( x e. dom F /\ y e. F ) ) -> ( x = |^| |^| y <-> y = <. x , ( F ` x ) >. ) )
59	58	ex 115	. 2 |- ( Fun F -> ( ( x e. dom F /\ y e. F ) -> ( x = |^| |^| y <-> y = <. x , ( F ` x ) >. ) ) )
60	3, 4, 6, 10, 59	en3d 6985	1 |- ( Fun F -> dom F ~~ F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803   C_ wss 3201   <.cop 3676   |^|cint 3933   class class class wbr 4093   X. cxp 4729   dom cdm 4731   Rel wrel 4736   Fun wfun 5327   ` cfv 5333   ~~ cen 6950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-en 6953

This theorem is referenced by:  fundmeng  7025