Theorem mapexi 6004

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 ∈ V
mapexi.2	⊢ B ∈ V

Assertion

Ref	Expression
mapexi	⊢ {f ∣ f:A–→B} ∈ 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 3220	. . . . . 6 ⊢ (f ∈ ( Funs ∩ (◡Image1st “ {A})) ↔ (f ∈ Funs ∧ f ∈ (◡Image1st “ {A})))
2		vex 2863	. . . . . . . 8 ⊢ f ∈ V
3	2	elfuns 5830	. . . . . . 7 ⊢ (f ∈ Funs ↔ Fun f)
4		elimasn 5020	. . . . . . . 8 ⊢ (f ∈ (◡Image1st “ {A}) ↔ ⟨A, f⟩ ∈ ◡Image1st )
5		df-br 4641	. . . . . . . 8 ⊢ (A◡Image1st f ↔ ⟨A, f⟩ ∈ ◡Image1st )
6		brcnv 4893	. . . . . . . . 9 ⊢ (A◡Image1st f ↔ fImage1st A)
7		mapexi.1	. . . . . . . . . . 11 ⊢ A ∈ V
8	2, 7	brimage 5794	. . . . . . . . . 10 ⊢ (fImage1st A ↔ A = (1st “ f))
9		dfdm4 5508	. . . . . . . . . . 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 ⊢ (fImage1st A ↔ dom f = A)
13	6, 12	bitri 240	. . . . . . . 8 ⊢ (A◡Image1st f ↔ dom f = A)
14	4, 5, 13	3bitr2i 264	. . . . . . 7 ⊢ (f ∈ (◡Image1st “ {A}) ↔ dom f = A)
15	3, 14	anbi12i 678	. . . . . 6 ⊢ ((f ∈ Funs ∧ f ∈ (◡Image1st “ {A})) ↔ (Fun f ∧ dom f = A))
16	1, 15	bitri 240	. . . . 5 ⊢ (f ∈ ( Funs ∩ (◡Image1st “ {A})) ↔ (Fun f ∧ dom f = A))
17		vex 2863	. . . . . . . . . 10 ⊢ x ∈ V
18	2, 17	brimage 5794	. . . . . . . . 9 ⊢ (fImage2nd x ↔ x = (2nd “ f))
19		brcnv 4893	. . . . . . . . 9 ⊢ (x◡Image2nd f ↔ fImage2nd x)
20		dfrn5 5509	. . . . . . . . . 10 ⊢ ran f = (2nd “ f)
21	20	eqeq2i 2363	. . . . . . . . 9 ⊢ (x = ran f ↔ x = (2nd “ f))
22	18, 19, 21	3bitr4i 268	. . . . . . . 8 ⊢ (x◡Image2nd f ↔ x = ran f)
23	22	rexbii 2640	. . . . . . 7 ⊢ (∃x ∈ ℘ Bx◡Image2nd f ↔ ∃x ∈ ℘ Bx = ran f)
24		elima 4755	. . . . . . 7 ⊢ (f ∈ (◡Image2nd “ ℘B) ↔ ∃x ∈ ℘ Bx◡Image2nd f)
25		risset 2662	. . . . . . 7 ⊢ (ran f ∈ ℘B ↔ ∃x ∈ ℘ Bx = ran f)
26	23, 24, 25	3bitr4i 268	. . . . . 6 ⊢ (f ∈ (◡Image2nd “ ℘B) ↔ ran f ∈ ℘B)
27	2	rnex 5108	. . . . . . 7 ⊢ ran f ∈ V
28	27	elpw 3729	. . . . . 6 ⊢ (ran f ∈ ℘B ↔ ran f ⊆ B)
29	26, 28	bitri 240	. . . . 5 ⊢ (f ∈ (◡Image2nd “ ℘B) ↔ ran f ⊆ B)
30	16, 29	anbi12i 678	. . . 4 ⊢ ((f ∈ ( Funs ∩ (◡Image1st “ {A})) ∧ f ∈ (◡Image2nd “ ℘B)) ↔ ((Fun f ∧ dom f = A) ∧ ran f ⊆ B))
31		elin 3220	. . . 4 ⊢ (f ∈ (( Funs ∩ (◡Image1st “ {A})) ∩ (◡Image2nd “ ℘B)) ↔ (f ∈ ( Funs ∩ (◡Image1st “ {A})) ∧ f ∈ (◡Image2nd “ ℘B)))
32		df-f 4792	. . . . 5 ⊢ (f:A–→B ↔ (f Fn A ∧ ran f ⊆ B))
33		df-fn 4791	. . . . . 6 ⊢ (f Fn A ↔ (Fun f ∧ dom f = A))
34	33	anbi1i 676	. . . . 5 ⊢ ((f Fn A ∧ ran f ⊆ B) ↔ ((Fun f ∧ dom f = A) ∧ ran f ⊆ B))
35	32, 34	bitri 240	. . . 4 ⊢ (f:A–→B ↔ ((Fun f ∧ dom f = A) ∧ ran f ⊆ B))
36	30, 31, 35	3bitr4i 268	. . 3 ⊢ (f ∈ (( Funs ∩ (◡Image1st “ {A})) ∩ (◡Image2nd “ ℘B)) ↔ f:A–→B)
37	36	abbi2i 2465	. 2 ⊢ (( Funs ∩ (◡Image1st “ {A})) ∩ (◡Image2nd “ ℘B)) = {f ∣ f:A–→B}
38		funsex 5829	. . . 4 ⊢ Funs ∈ V
39		1stex 4740	. . . . . . 7 ⊢ 1st ∈ V
40	39	imageex 5802	. . . . . 6 ⊢ Image1st ∈ V
41	40	cnvex 5103	. . . . 5 ⊢ ◡Image1st ∈ V
42		snex 4112	. . . . 5 ⊢ {A} ∈ V
43	41, 42	imaex 4748	. . . 4 ⊢ (◡Image1st “ {A}) ∈ V
44	38, 43	inex 4106	. . 3 ⊢ ( Funs ∩ (◡Image1st “ {A})) ∈ V
45		2ndex 5113	. . . . . 6 ⊢ 2nd ∈ V
46	45	imageex 5802	. . . . 5 ⊢ Image2nd ∈ V
47	46	cnvex 5103	. . . 4 ⊢ ◡Image2nd ∈ V
48		mapexi.2	. . . . 5 ⊢ B ∈ V
49	48	pwex 4330	. . . 4 ⊢ ℘B ∈ V
50	47, 49	imaex 4748	. . 3 ⊢ (◡Image2nd “ ℘B) ∈ V
51	44, 50	inex 4106	. 2 ⊢ (( Funs ∩ (◡Image1st “ {A})) ∩ (◡Image2nd “ ℘B)) ∈ V
52	37, 51	eqeltrri 2424	1 ⊢ {f ∣ f:A–→B} ∈ V

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860   ∩ cin 3209   ⊆ wss 3258  ℘cpw 3723  {csn 3738  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   “ cima 4723  ^◡ccnv 4772  dom cdm 4773  ran crn 4774  Fun wfun 4776   Fn wfn 4777  –→wf 4778  2^nd c2nd 4784  Imagecimage 5754   Funs cfuns 5760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761

This theorem is referenced by:  mapex  6007  fnmap  6008