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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
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 5857 . . . . . . 7 Funs
4 elimasn 5034 . . . . . . . 8 Image Image
5 df-br 4632 . . . . . . . 8 Image Image
6 brcnv 4904 . . . . . . . . 9 Image Image
7 mapexi.1 . . . . . . . . . . 11
82, 7brimage 5822 . . . . . . . . . 10 Image
9 dfdm4 5548 . . . . . . . . . . 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 5822 . . . . . . . . 9 Image
19 brcnv 4904 . . . . . . . . 9 Image Image
20 dfrn5 5549 . . . . . . . . . 10
2120eqeq2i 2363 . . . . . . . . 9
2218, 19, 213bitr4i 268 . . . . . . . 8 Image
2322rexbii 2639 . . . . . . 7 Image
24 elima 4746 . . . . . . 7 Image Image
25 risset 2661 . . . . . . 7
2623, 24, 253bitr4i 268 . . . . . 6 Image
272rnex 5140 . . . . . . 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 4785 . . . . 5
33 df-fn 4784 . . . . . 6
3433anbi1i 676 . . . . 5
3532, 34bitri 240 . . . 4
3630, 31, 353bitr4i 268 . . 3 Funs Image Image
3736abbi2i 2464 . 2 Funs Image Image
38 funsex 5856 . . . 4 Funs
39 1stex 4731 . . . . . . 7
4039imageex 5830 . . . . . 6 Image
4140cnvex 5135 . . . . 5 Image
42 snex 4111 . . . . 5
4341, 42imaex 4739 . . . 4 Image
4438, 43inex 4105 . . 3 Funs Image
45 2ndex 5145 . . . . . 6
4645imageex 5830 . . . . 5 Image
4746cnvex 5135 . . . 4 Image
48 mapexi.2 . . . . 5
4948pwex 4329 . . . 4
5047, 49imaex 4739 . . 3 Image
5144, 50inex 4105 . 2 Funs Image Image
5237, 51eqeltrri 2424 1
Colors of variables: wff set 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 4631  c1st 4709  cima 4714  ccnv 4763   cdm 4764   crn 4765   wfun 4768   wfn 4769  wf 4770  c2nd 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
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