Theorem fundmen 6922

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 4964	. . 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 5715	. . 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 5307	. . 3 |- ( Fun F -> Rel F )
8		elreldm 4923	. . . 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 4700	. . . . . . . . 9 |- ( Rel F <-> F C_ ( _V X. _V ) )
12	7, 11	sylib 122	. . . . . . . 8 |- ( Fun F -> F C_ ( _V X. _V ) )
13	12	sselda 3201	. . . . . . 7 |- ( ( Fun F /\ y e. F ) -> y e. ( _V X. _V ) )
14		elvv 4755	. . . . . . 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 3902	. . . . . . . . . . . . . . . . 17 |- ( y = <. z , w >. -> |^| y = |^| <. z , w >. )
17	16	inteqd 3904	. . . . . . . . . . . . . . . 16 |- ( y = <. z , w >. -> |^| |^| y = |^| |^| <. z , w >. )
18		vex 2779	. . . . . . . . . . . . . . . . 17 |- z e. _V
19		vex 2779	. . . . . . . . . . . . . . . . 17 |- w e. _V
20	18, 19	op1stb 4543	. . . . . . . . . . . . . . . 16 |- |^| |^| <. z , w >. = z
21	17, 20	eqtrdi 2256	. . . . . . . . . . . . . . 15 |- ( y = <. z , w >. -> |^| |^| y = z )
22		eqeq1 2214	. . . . . . . . . . . . . . 15 |- ( x = |^| |^| y -> ( x = z <-> |^| |^| y = z ) )
23	21, 22	imbitrrid 156	. . . . . . . . . . . . . 14 |- ( x = |^| |^| y -> ( y = <. z , w >. -> x = z ) )
24		opeq1 3833	. . . . . . . . . . . . . 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 2217	. . . . . . . . . . . . . 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 2276	. . . . . . . . . . . . . . 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 5641	. . . . . . . . . . . . . . 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 3840	. . . . . . . . 9 |- ( ( ( Fun F /\ y e. F ) /\ ( y = <. z , w >. /\ x = |^| |^| y ) ) -> <. x , ( F ` x ) >. = <. x , w >. )
42	32, 41	eqtr4d 2243	. . . . . . . 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 1922	. . . . . 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 3902	. . . . . . . . 9 |- ( y = <. x , ( F ` x ) >. -> |^| y = |^| <. x , ( F ` x ) >. )
48	47	inteqd 3904	. . . . . . . 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 2779	. . . . . . . . 9 |- x e. _V
51		funfvex 5616	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
52		op1stbg 4544	. . . . . . . . 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 2241	. . . . . 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 6883	1 |- ( Fun F -> dom F ~~ F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2178   _Vcvv 2776   C_ wss 3174   <.cop 3646   |^|cint 3899   class class class wbr 4059   X. cxp 4691   dom cdm 4693   Rel wrel 4698   Fun wfun 5284   ` cfv 5290   ~~ cen 6848

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-en 6851

This theorem is referenced by:  fundmeng  6923