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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
mapexi.2
Assertion
Ref Expression
mapexi
Distinct variable groups:   ,   ,

Proof of Theorem mapexi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3219 . . . . . 6 Funs Image Funs Image
2 vex 2862 . . . . . . . 8
32elfuns 5829 . . . . . . 7 Funs
4 elimasn 5019 . . . . . . . 8 Image Image
5 df-br 4640 . . . . . . . 8 Image Image
6 brcnv 4892 . . . . . . . . 9 Image Image
7 mapexi.1 . . . . . . . . . . 11
82, 7brimage 5793 . . . . . . . . . 10 Image
9 dfdm4 5507 . . . . . . . . . . 11
109eqeq2i 2363 . . . . . . . . . 10
11 eqcom 2355 . . . . . . . . . 10
128, 10, 113bitr2i 264 . . . . . . . . 9 Image
136, 12bitri 240 . . . . . . . 8 Image
144, 5, 133bitr2i 264 . . . . . . 7 Image
153, 14anbi12i 678 . . . . . 6 Funs Image
161, 15bitri 240 . . . . 5 Funs Image
17 vex 2862 . . . . . . . . . 10
182, 17brimage 5793 . . . . . . . . 9 Image
19 brcnv 4892 . . . . . . . . 9 Image Image
20 dfrn5 5508 . . . . . . . . . 10
2120eqeq2i 2363 . . . . . . . . 9
2218, 19, 213bitr4i 268 . . . . . . . 8 Image
2322rexbii 2639 . . . . . . 7 Image
24 elima 4754 . . . . . . 7 Image Image
25 risset 2661 . . . . . . 7
2623, 24, 253bitr4i 268 . . . . . 6 Image
272rnex 5107 . . . . . . 7
2827elpw 3728 . . . . . 6
2926, 28bitri 240 . . . . 5 Image
3016, 29anbi12i 678 . . . 4 Funs Image Image
31 elin 3219 . . . 4 Funs Image Image Funs Image Image
32 df-f 4791 . . . . 5
33 df-fn 4790 . . . . . 6
3433anbi1i 676 . . . . 5
3532, 34bitri 240 . . . 4
3630, 31, 353bitr4i 268 . . 3 Funs Image Image
3736abbi2i 2464 . 2 Funs Image Image
38 funsex 5828 . . . 4 Funs
39 1stex 4739 . . . . . . 7
4039imageex 5801 . . . . . 6 Image
4140cnvex 5102 . . . . 5 Image
42 snex 4111 . . . . 5
4341, 42imaex 4747 . . . 4 Image
4438, 43inex 4105 . . 3 Funs Image
45 2ndex 5112 . . . . . 6
4645imageex 5801 . . . . 5 Image
4746cnvex 5102 . . . 4 Image
48 mapexi.2 . . . . 5
4948pwex 4329 . . . 4
5047, 49imaex 4747 . . 3 Image
5144, 50inex 4105 . 2 Funs Image Image
5237, 51eqeltrri 2424 1
 Colors of variables: wff setvar class Syntax hints:   wa 358   wceq 1642   wcel 1710  cab 2339  wrex 2615  cvv 2859   cin 3208   wss 3257  cpw 3722  csn 3737  cop 4561   class class class wbr 4639  c1st 4717  cima 4722  ccnv 4771   cdm 4772   crn 4773   wfun 4775   wfn 4776  wf 4777  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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