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Theorem composefn 5818
 Description: The compose function is a function over the universe. (Contributed by Scott Fenton, 19-Apr-2021.)
Assertion
Ref Expression
composefn Compose

Proof of Theorem composefn
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . . 6
2 vex 2862 . . . . . 6
31, 2coex 4750 . . . . 5
43eueq1 3009 . . . 4
54a1i 10 . . 3
65fnoprab 5586 . 2
7 df-compose 5748 . . . 4 Compose
8 df-mpt2 5654 . . . 4
97, 8eqtri 2373 . . 3 Compose
10 xpvv 4843 . . . 4
11 df-xp 4784 . . . 4
1210, 11eqtr3i 2375 . . 3
13 fneq1 5173 . . . 4 Compose Compose
14 fneq2 5174 . . . 4
1513, 14sylan9bb 680 . . 3 Compose Compose
169, 12, 15mp2an 653 . 2 Compose
176, 16mpbir 200 1 Compose
 Colors of variables: wff setvar class Syntax hints:   wb 176   wa 358   wceq 1642   wcel 1710  weu 2204  cvv 2859  copab 4622   ccom 4721   cxp 4770   wfn 4776  coprab 5527   cmpt2 5653   Compose ccompose 5747 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 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-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-oprab 5528  df-mpt2 5654  df-compose 5748 This theorem is referenced by:  brcomposeg  5819
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