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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 Fn V

Proof of Theorem composefn
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . . 6 x V
2 vex 2862 . . . . . 6 y V
31, 2coex 4750 . . . . 5 (x y) V
43eueq1 3009 . . . 4 ∃!z z = (x y)
54a1i 10 . . 3 ((x V y V) → ∃!z z = (x y))
65fnoprab 5586 . 2 {x, y, z ((x V y V) z = (x y))} Fn {x, y (x V y V)}
7 df-compose 5748 . . . 4 Compose = (x V, y V (x y))
8 df-mpt2 5654 . . . 4 (x V, y V (x y)) = {x, y, z ((x V y V) z = (x y))}
97, 8eqtri 2373 . . 3 Compose = {x, y, z ((x V y V) z = (x y))}
10 xpvv 4843 . . . 4 (V × V) = V
11 df-xp 4784 . . . 4 (V × V) = {x, y (x V y V)}
1210, 11eqtr3i 2375 . . 3 V = {x, y (x V y V)}
13 fneq1 5173 . . . 4 ( Compose = {x, y, z ((x V y V) z = (x y))} → ( Compose Fn V ↔ {x, y, z ((x V y V) z = (x y))} Fn V))
14 fneq2 5174 . . . 4 (V = {x, y (x V y V)} → ({x, y, z ((x V y V) z = (x y))} Fn V ↔ {x, y, z ((x V y V) z = (x y))} Fn {x, y (x V y V)}))
1513, 14sylan9bb 680 . . 3 (( Compose = {x, y, z ((x V y V) z = (x y))} V = {x, y (x V y V)}) → ( Compose Fn V ↔ {x, y, z ((x V y V) z = (x y))} Fn {x, y (x V y V)}))
169, 12, 15mp2an 653 . 2 ( Compose Fn V ↔ {x, y, z ((x V y V) z = (x y))} Fn {x, y (x V y V)})
176, 16mpbir 200 1 Compose Fn V
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2859  {copab 4622   ∘ ccom 4721   × cxp 4770   Fn 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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