Theorem fundmen 6043

Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by SF, 23-Feb-2015.)

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 a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssv 3291	. . . . . 6 |- F C_ _V
2		1stfo 5505	. . . . . . 7 |- 1st:_V-onto->_V
3		fofn 5271	. . . . . . 7 |- (1st:_V-onto->_V -> 1st Fn _V)
4		fnssresb 5195	. . . . . . 7 |- (1st Fn _V -> ((1st |` F) Fn F <-> F C_ _V))
5	2, 3, 4	mp2b 9	. . . . . 6 |- ((1st |` F) Fn F <-> F C_ _V)
6	1, 5	mpbir 200	. . . . 5 |- (1st |` F) Fn F
7	6	a1i 10	. . . 4 |- (Fun F -> (1st |` F) Fn F)
8		brcnv 4892	. . . . . . . . . . 11 |- (x`'(1st |` F)y <-> y(1st |` F)x)
9		brres 4949	. . . . . . . . . . 11 |- (y(1st |` F)x <-> (y1stx /\ y e. F))
10		vex 2862	. . . . . . . . . . . . . 14 |- x e. _V
11	10	br1st 4858	. . . . . . . . . . . . 13 |- (y1stx <-> E.a y = <.x, a>.)
12	11	anbi1i 676	. . . . . . . . . . . 12 |- ((y1stx /\ y e. F) <-> (E.a y = <.x, a>. /\ y e. F))
13		19.41v 1901	. . . . . . . . . . . 12 |- (E.a(y = <.x, a>. /\ y e. F) <-> (E.a y = <.x, a>. /\ y e. F))
14	12, 13	bitr4i 243	. . . . . . . . . . 11 |- ((y1stx /\ y e. F) <-> E.a(y = <.x, a>. /\ y e. F))
15	8, 9, 14	3bitri 262	. . . . . . . . . 10 |- (x`'(1st |` F)y <-> E.a(y = <.x, a>. /\ y e. F))
16		brcnv 4892	. . . . . . . . . . . 12 |- (x`'(1st |` F)z <-> z(1st |` F)x)
17		brres 4949	. . . . . . . . . . . 12 |- (z(1st |` F)x <-> (z1stx /\ z e. F))
18	10	br1st 4858	. . . . . . . . . . . . 13 |- (z1stx <-> E.b z = <.x, b>.)
19	18	anbi1i 676	. . . . . . . . . . . 12 |- ((z1stx /\ z e. F) <-> (E.b z = <.x, b>. /\ z e. F))
20	16, 17, 19	3bitri 262	. . . . . . . . . . 11 |- (x`'(1st |` F)z <-> (E.b z = <.x, b>. /\ z e. F))
21		19.41v 1901	. . . . . . . . . . 11 |- (E.b(z = <.x, b>. /\ z e. F) <-> (E.b z = <.x, b>. /\ z e. F))
22	20, 21	bitr4i 243	. . . . . . . . . 10 |- (x`'(1st |` F)z <-> E.b(z = <.x, b>. /\ z e. F))
23	15, 22	anbi12i 678	. . . . . . . . 9 |- ((x`'(1st |` F)y /\ x`'(1st |` F)z) <-> (E.a(y = <.x, a>. /\ y e. F) /\ E.b(z = <.x, b>. /\ z e. F)))
24		eeanv 1913	. . . . . . . . 9 |- (E.aE.b((y = <.x, a>. /\ y e. F) /\ (z = <.x, b>. /\ z e. F)) <-> (E.a(y = <.x, a>. /\ y e. F) /\ E.b(z = <.x, b>. /\ z e. F)))
25	23, 24	bitr4i 243	. . . . . . . 8 |- ((x`'(1st |` F)y /\ x`'(1st |` F)z) <-> E.aE.b((y = <.x, a>. /\ y e. F) /\ (z = <.x, b>. /\ z e. F)))
26		an4 797	. . . . . . . . . 10 |- (((y = <.x, a>. /\ y e. F) /\ (z = <.x, b>. /\ z e. F)) <-> ((y = <.x, a>. /\ z = <.x, b>.) /\ (y e. F /\ z e. F)))
27		dffun4 5121	. . . . . . . . . . . . 13 |- (Fun F <-> A.xA.aA.b((<.x, a>. e. F /\ <.x, b>. e. F) -> a = b))
28		sp 1747	. . . . . . . . . . . . . . 15 |- (A.b((<.x, a>. e. F /\ <.x, b>. e. F) -> a = b) -> ((<.x, a>. e. F /\ <.x, b>. e. F) -> a = b))
29	28	sps 1754	. . . . . . . . . . . . . 14 |- (A.aA.b((<.x, a>. e. F /\ <.x, b>. e. F) -> a = b) -> ((<.x, a>. e. F /\ <.x, b>. e. F) -> a = b))
30	29	sps 1754	. . . . . . . . . . . . 13 |- (A.xA.aA.b((<.x, a>. e. F /\ <.x, b>. e. F) -> a = b) -> ((<.x, a>. e. F /\ <.x, b>. e. F) -> a = b))
31	27, 30	sylbi 187	. . . . . . . . . . . 12 |- (Fun F -> ((<.x, a>. e. F /\ <.x, b>. e. F) -> a = b))
32		opeq2 4579	. . . . . . . . . . . 12 |- (a = b -> <.x, a>. = <.x, b>.)
33	31, 32	syl6 29	. . . . . . . . . . 11 |- (Fun F -> ((<.x, a>. e. F /\ <.x, b>. e. F) -> <.x, a>. = <.x, b>.))
34		eleq1 2413	. . . . . . . . . . . . . . 15 |- (y = <.x, a>. -> (y e. F <-> <.x, a>. e. F))
35		eleq1 2413	. . . . . . . . . . . . . . 15 |- (z = <.x, b>. -> (z e. F <-> <.x, b>. e. F))
36	34, 35	bi2anan9 843	. . . . . . . . . . . . . 14 |- ((y = <.x, a>. /\ z = <.x, b>.) -> ((y e. F /\ z e. F) <-> (<.x, a>. e. F /\ <.x, b>. e. F)))
37		eqeq12 2365	. . . . . . . . . . . . . 14 |- ((y = <.x, a>. /\ z = <.x, b>.) -> (y = z <-> <.x, a>. = <.x, b>.))
38	36, 37	imbi12d 311	. . . . . . . . . . . . 13 |- ((y = <.x, a>. /\ z = <.x, b>.) -> (((y e. F /\ z e. F) -> y = z) <-> ((<.x, a>. e. F /\ <.x, b>. e. F) -> <.x, a>. = <.x, b>.)))
39	38	biimprcd 216	. . . . . . . . . . . 12 |- (((<.x, a>. e. F /\ <.x, b>. e. F) -> <.x, a>. = <.x, b>.) -> ((y = <.x, a>. /\ z = <.x, b>.) -> ((y e. F /\ z e. F) -> y = z)))
40	39	imp3a 420	. . . . . . . . . . 11 |- (((<.x, a>. e. F /\ <.x, b>. e. F) -> <.x, a>. = <.x, b>.) -> (((y = <.x, a>. /\ z = <.x, b>.) /\ (y e. F /\ z e. F)) -> y = z))
41	33, 40	syl 15	. . . . . . . . . 10 |- (Fun F -> (((y = <.x, a>. /\ z = <.x, b>.) /\ (y e. F /\ z e. F)) -> y = z))
42	26, 41	syl5bi 208	. . . . . . . . 9 |- (Fun F -> (((y = <.x, a>. /\ y e. F) /\ (z = <.x, b>. /\ z e. F)) -> y = z))
43	42	exlimdvv 1637	. . . . . . . 8 |- (Fun F -> (E.aE.b((y = <.x, a>. /\ y e. F) /\ (z = <.x, b>. /\ z e. F)) -> y = z))
44	25, 43	syl5bi 208	. . . . . . 7 |- (Fun F -> ((x`'(1st |` F)y /\ x`'(1st |` F)z) -> y = z))
45	44	alrimiv 1631	. . . . . 6 |- (Fun F -> A.z((x`'(1st |` F)y /\ x`'(1st |` F)z) -> y = z))
46	45	alrimivv 1632	. . . . 5 |- (Fun F -> A.xA.yA.z((x`'(1st |` F)y /\ x`'(1st |` F)z) -> y = z))
47		dffun2 5119	. . . . 5 |- (Fun `'(1st |` F) <-> A.xA.yA.z((x`'(1st |` F)y /\ x`'(1st |` F)z) -> y = z))
48	46, 47	sylibr 203	. . . 4 |- (Fun F -> Fun `'(1st |` F))
49		dfdm4 5507	. . . . . 6 |- dom F = (1st"F)
50		dfima3 4951	. . . . . 6 |- (1st"F) = ran (1st |` F)
51	49, 50	eqtr2i 2374	. . . . 5 |- ran (1st |` F) = dom F
52	51	a1i 10	. . . 4 |- (Fun F -> ran (1st |` F) = dom F)
53		dff1o2 5291	. . . 4 |- ((1st |` F):F-1-1-onto->dom F <-> ((1st |` F) Fn F /\ Fun `'(1st |` F) /\ ran (1st |` F) = dom F))
54	7, 48, 52, 53	syl3anbrc 1136	. . 3 |- (Fun F -> (1st |` F):F-1-1-onto->dom F)
55		1stex 4739	. . . . 5 |- 1st e. _V
56		fundmen.1	. . . . 5 |- F e. _V
57	55, 56	resex 5117	. . . 4 |- (1st |` F) e. _V
58	57	f1oen 6033	. . 3 |- ((1st |` F):F-1-1-onto->dom F -> F ~~ dom F)
59	54, 58	syl 15	. 2 |- (Fun F -> F ~~ dom F)
60		ensym 6037	. 2 |- (F ~~ dom F <-> dom F ~~ F)
61	59, 60	sylib 188	1 |- (Fun F -> dom F ~~ F)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2859   C_ wss 3257  <.cop 4561   class class class wbr 4639  1stc1st 4717  "cima 4722  `'ccnv 4771  dom cdm 4772  ran crn 4773   |` cres 4774  Fun wfun 4775   Fn wfn 4776  -onto->wfo 4779  -1-1-onto->wf1o 4780   ~~ cen 6028

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-en 6029

This theorem is referenced by:  fundmeng  6044  xpsnen  6049