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Theorem mapexi 6003
 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.
StepHypRef Expression
1 elin 3219 . . . . . 6 (f ( Funs ∩ (Image1st “ {A})) ↔ (f Funs f (Image1st “ {A})))
2 vex 2862 . . . . . . . 8 f V
32elfuns 5829 . . . . . . 7 (f Funs ↔ Fun f)
4 elimasn 5019 . . . . . . . 8 (f (Image1st “ {A}) ↔ A, f Image1st )
5 df-br 4640 . . . . . . . 8 (AImage1st fA, f Image1st )
6 brcnv 4892 . . . . . . . . 9 (AImage1st ffImage1st A)
7 mapexi.1 . . . . . . . . . . 11 A V
82, 7brimage 5793 . . . . . . . . . 10 (fImage1st AA = (1stf))
9 dfdm4 5507 . . . . . . . . . . 11 dom f = (1stf)
109eqeq2i 2363 . . . . . . . . . 10 (A = dom fA = (1stf))
11 eqcom 2355 . . . . . . . . . 10 (A = dom f ↔ dom f = A)
128, 10, 113bitr2i 264 . . . . . . . . 9 (fImage1st A ↔ dom f = A)
136, 12bitri 240 . . . . . . . 8 (AImage1st f ↔ dom f = A)
144, 5, 133bitr2i 264 . . . . . . 7 (f (Image1st “ {A}) ↔ dom f = A)
153, 14anbi12i 678 . . . . . 6 ((f Funs f (Image1st “ {A})) ↔ (Fun f dom f = A))
161, 15bitri 240 . . . . 5 (f ( Funs ∩ (Image1st “ {A})) ↔ (Fun f dom f = A))
17 vex 2862 . . . . . . . . . 10 x V
182, 17brimage 5793 . . . . . . . . 9 (fImage2nd xx = (2ndf))
19 brcnv 4892 . . . . . . . . 9 (xImage2nd ffImage2nd x)
20 dfrn5 5508 . . . . . . . . . 10 ran f = (2ndf)
2120eqeq2i 2363 . . . . . . . . 9 (x = ran fx = (2ndf))
2218, 19, 213bitr4i 268 . . . . . . . 8 (xImage2nd fx = ran f)
2322rexbii 2639 . . . . . . 7 (x BxImage2nd fx Bx = ran f)
24 elima 4754 . . . . . . 7 (f (Image2ndB) ↔ x BxImage2nd f)
25 risset 2661 . . . . . . 7 (ran f Bx Bx = ran f)
2623, 24, 253bitr4i 268 . . . . . 6 (f (Image2ndB) ↔ ran f B)
272rnex 5107 . . . . . . 7 ran f V
2827elpw 3728 . . . . . 6 (ran f B ↔ ran f B)
2926, 28bitri 240 . . . . 5 (f (Image2ndB) ↔ ran f B)
3016, 29anbi12i 678 . . . 4 ((f ( Funs ∩ (Image1st “ {A})) f (Image2ndB)) ↔ ((Fun f dom f = A) ran f B))
31 elin 3219 . . . 4 (f (( Funs ∩ (Image1st “ {A})) ∩ (Image2ndB)) ↔ (f ( Funs ∩ (Image1st “ {A})) f (Image2ndB)))
32 df-f 4791 . . . . 5 (f:A–→B ↔ (f Fn A ran f B))
33 df-fn 4790 . . . . . 6 (f Fn A ↔ (Fun f dom f = A))
3433anbi1i 676 . . . . 5 ((f Fn A ran f B) ↔ ((Fun f dom f = A) ran f B))
3532, 34bitri 240 . . . 4 (f:A–→B ↔ ((Fun f dom f = A) ran f B))
3630, 31, 353bitr4i 268 . . 3 (f (( Funs ∩ (Image1st “ {A})) ∩ (Image2ndB)) ↔ f:A–→B)
3736abbi2i 2464 . 2 (( Funs ∩ (Image1st “ {A})) ∩ (Image2ndB)) = {f f:A–→B}
38 funsex 5828 . . . 4 Funs V
39 1stex 4739 . . . . . . 7 1st V
4039imageex 5801 . . . . . 6 Image1st V
4140cnvex 5102 . . . . 5 Image1st V
42 snex 4111 . . . . 5 {A} V
4341, 42imaex 4747 . . . 4 (Image1st “ {A}) V
4438, 43inex 4105 . . 3 ( Funs ∩ (Image1st “ {A})) V
45 2ndex 5112 . . . . . 6 2nd V
4645imageex 5801 . . . . 5 Image2nd V
4746cnvex 5102 . . . 4 Image2nd V
48 mapexi.2 . . . . 5 B V
4948pwex 4329 . . . 4 B V
5047, 49imaex 4747 . . 3 (Image2ndB) V
5144, 50inex 4105 . 2 (( Funs ∩ (Image1st “ {A})) ∩ (Image2ndB)) V
5237, 51eqeltrri 2424 1 {f f:A–→B} V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859   ∩ cin 3208   ⊆ wss 3257  ℘cpw 3722  {csn 3737  ⟨cop 4561   class class class wbr 4639  1st c1st 4717   “ cima 4722  ◡ccnv 4771  dom cdm 4772  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –→wf 4777  2nd c2nd 4783  Imagecimage 5753   Funs cfuns 5759 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-sset 4725  df-co 4726  df-ima 4727  df-si 4728  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-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760 This theorem is referenced by:  mapex  6006  fnmap  6007
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