Theorem mapexi 6023

Description: The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by set.mm contributors, 25-Feb-2015.)

Hypotheses

Ref	Expression
mapexi.1	|- A e. _V
mapexi.2	|- B e. _V

Assertion

Ref	Expression
mapexi	|- {f | f:A-->B} e. _V

Distinct variable groups:   A,f   B,f

Proof of Theorem mapexi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3219	. . . . . 6 |- (f e. ( Funs i^i (`'Image1st"{A})) <-> (f e. Funs /\ f e. (`'Image1st"{A})))
2		vex 2862	. . . . . . . 8 |- f e. _V
3	2	elfuns 5857	. . . . . . 7 |- (f e. Funs <-> Fun f)
4		elimasn 5034	. . . . . . . 8 |- (f e. (`'Image1st"{A}) <-> <.A, f>. e. `'Image1st)
5		df-br 4632	. . . . . . . 8 |- (A`'Image1stf <-> <.A, f>. e. `'Image1st)
6		brcnv 4904	. . . . . . . . 9 |- (A`'Image1stf <-> fImage1stA)
7		mapexi.1	. . . . . . . . . . 11 |- A e. _V
8	2, 7	brimage 5822	. . . . . . . . . 10 |- (fImage1stA <-> A = (1st"f))
9		dfdm4 5548	. . . . . . . . . . 11 |- dom f = (1st"f)
10	9	eqeq2i 2363	. . . . . . . . . 10 |- (A = dom f <-> A = (1st"f))
11		eqcom 2355	. . . . . . . . . 10 |- (A = dom f <-> dom f = A)
12	8, 10, 11	3bitr2i 264	. . . . . . . . 9 |- (fImage1stA <-> dom f = A)
13	6, 12	bitri 240	. . . . . . . 8 |- (A`'Image1stf <-> dom f = A)
14	4, 5, 13	3bitr2i 264	. . . . . . 7 |- (f e. (`'Image1st"{A}) <-> dom f = A)
15	3, 14	anbi12i 678	. . . . . 6 |- ((f e. Funs /\ f e. (`'Image1st"{A})) <-> (Fun f /\ dom f = A))
16	1, 15	bitri 240	. . . . 5 |- (f e. ( Funs i^i (`'Image1st"{A})) <-> (Fun f /\ dom f = A))
17		vex 2862	. . . . . . . . . 10 |- x e. _V
18	2, 17	brimage 5822	. . . . . . . . 9 |- (fImage2ndx <-> x = (2nd"f))
19		brcnv 4904	. . . . . . . . 9 |- (x`'Image2ndf <-> fImage2ndx)
20		dfrn5 5549	. . . . . . . . . 10 |- ran f = (2nd"f)
21	20	eqeq2i 2363	. . . . . . . . 9 |- (x = ran f <-> x = (2nd"f))
22	18, 19, 21	3bitr4i 268	. . . . . . . 8 |- (x`'Image2ndf <-> x = ran f)
23	22	rexbii 2639	. . . . . . 7 |- (E.x e. ~P Bx`'Image2ndf <-> E.x e. ~P Bx = ran f)
24		elima 4746	. . . . . . 7 |- (f e. (`'Image2nd"~PB) <-> E.x e. ~P Bx`'Image2ndf)
25		risset 2661	. . . . . . 7 |- (ran f e. ~PB <-> E.x e. ~P Bx = ran f)
26	23, 24, 25	3bitr4i 268	. . . . . 6 |- (f e. (`'Image2nd"~PB) <-> ran f e. ~PB)
27	2	rnex 5140	. . . . . . 7 |- ran f e. _V
28	27	elpw 3728	. . . . . 6 |- (ran f e. ~PB <-> ran f C_ B)
29	26, 28	bitri 240	. . . . 5 |- (f e. (`'Image2nd"~PB) <-> ran f C_ B)
30	16, 29	anbi12i 678	. . . 4 |- ((f e. ( Funs i^i (`'Image1st"{A})) /\ f e. (`'Image2nd"~PB)) <-> ((Fun f /\ dom f = A) /\ ran f C_ B))
31		elin 3219	. . . 4 |- (f e. (( Funs i^i (`'Image1st"{A})) i^i (`'Image2nd"~PB)) <-> (f e. ( Funs i^i (`'Image1st"{A})) /\ f e. (`'Image2nd"~PB)))
32		df-f 4785	. . . . 5 |- (f:A-->B <-> (f Fn A /\ ran f C_ B))
33		df-fn 4784	. . . . . 6 |- (f Fn A <-> (Fun f /\ dom f = A))
34	33	anbi1i 676	. . . . 5 |- ((f Fn A /\ ran f C_ B) <-> ((Fun f /\ dom f = A) /\ ran f C_ B))
35	32, 34	bitri 240	. . . 4 |- (f:A-->B <-> ((Fun f /\ dom f = A) /\ ran f C_ B))
36	30, 31, 35	3bitr4i 268	. . 3 |- (f e. (( Funs i^i (`'Image1st"{A})) i^i (`'Image2nd"~PB)) <-> f:A-->B)
37	36	abbi2i 2464	. 2 |- (( Funs i^i (`'Image1st"{A})) i^i (`'Image2nd"~PB)) = {f | f:A-->B}
38		funsex 5856	. . . 4 |- Funs e. _V
39		1stex 4731	. . . . . . 7 |- 1st e. _V
40	39	imageex 5830	. . . . . 6 |- Image1st e. _V
41	40	cnvex 5135	. . . . 5 |- `'Image1st e. _V
42		snex 4111	. . . . 5 |- {A} e. _V
43	41, 42	imaex 4739	. . . 4 |- (`'Image1st"{A}) e. _V
44	38, 43	inex 4105	. . 3 |- ( Funs i^i (`'Image1st"{A})) e. _V
45		2ndex 5145	. . . . . 6 |- 2nd e. _V
46	45	imageex 5830	. . . . 5 |- Image2nd e. _V
47	46	cnvex 5135	. . . 4 |- `'Image2nd e. _V
48		mapexi.2	. . . . 5 |- B e. _V
49	48	pwex 4329	. . . 4 |- ~PB e. _V
50	47, 49	imaex 4739	. . 3 |- (`'Image2nd"~PB) e. _V
51	44, 50	inex 4105	. 2 |- (( Funs i^i (`'Image1st"{A})) i^i (`'Image2nd"~PB)) e. _V
52	37, 51	eqeltrri 2424	1 |- {f | f:A-->B} e. _V

Syntax hints:   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2615  _Vcvv 2859   i^i cin 3208   C_ wss 3257  ~Pcpw 3722  {csn 3737  <.cop 4561   class class class wbr 4631  1stc1st 4709  "cima 4714  `'ccnv 4763  dom cdm 4764  ran crn 4765  Fun wfun 4768   Fn wfn 4769  -->wf 4770  2ndc2nd 4776  Imagecimage 5774   Funs cfuns 5782

This theorem is referenced by:  mapex  6026  fnmap  6027

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 4439  df-ltfin 4440  df-ncfin 4441  df-tfin 4442  df-evenfin 4443  df-oddfin 4444  df-sfin 4445  df-spfin 4446  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4615  df-br 4632  df-1st 4715  df-swap 4716  df-sset 4717  df-co 4718  df-ima 4719  df-si 4720  df-id 4759  df-xp 4777  df-rel 4778  df-cnv 4779  df-rn 4780  df-dm 4781  df-res 4782  df-fun 4783  df-fn 4784  df-f 4785  df-fo 4787  df-fv 4789  df-2nd 4791  df-txp 5787  df-ins2 5793  df-ins3 5794  df-image 5795  df-ins4 5796  df-si3 5797  df-funs 5798