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Theorem funco 5142
 Description: The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
funco ((Fun F Fun G) → Fun (F G))

Proof of Theorem funco
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmo 5125 . . . . 5 (Fun G∃*z xGz)
2 funmo 5125 . . . . . 6 (Fun F∃*y zFy)
32alrimiv 1631 . . . . 5 (Fun Fz∃*y zFy)
4 moexexv 2274 . . . . 5 ((∃*z xGz z∃*y zFy) → ∃*yz(xGz zFy))
51, 3, 4syl2anr 464 . . . 4 ((Fun F Fun G) → ∃*yz(xGz zFy))
65alrimiv 1631 . . 3 ((Fun F Fun G) → x∃*yz(xGz zFy))
7 funopab 5139 . . 3 (Fun {x, y z(xGz zFy)} ↔ x∃*yz(xGz zFy))
86, 7sylibr 203 . 2 ((Fun F Fun G) → Fun {x, y z(xGz zFy)})
9 df-co 4726 . . 3 (F G) = {x, y z(xGz zFy)}
109funeqi 5128 . 2 (Fun (F G) ↔ Fun {x, y z(xGz zFy)})
118, 10sylibr 203 1 ((Fun F Fun G) → Fun (F G))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃*wmo 2205  {copab 4622   class class class wbr 4639   ∘ ccom 4721  Fun wfun 4775 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-co 4726  df-id 4767  df-cnv 4785  df-fun 4789 This theorem is referenced by:  fnco  5191  f1co  5264  sbthlem3  6205
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