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Theorem funco 4883
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 (𝐹𝐺))

Proof of Theorem funco
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmcoss 4544 . . . . 5 dom (𝐹𝐺) ⊆ dom 𝐺
2 funmo 4860 . . . . . . . . . 10 (Fun 𝐹∃*y z𝐹y)
32alrimiv 1751 . . . . . . . . 9 (Fun 𝐹z∃*y z𝐹y)
43ralrimivw 2387 . . . . . . . 8 (Fun 𝐹x dom 𝐺z∃*y z𝐹y)
5 dffun8 4872 . . . . . . . . 9 (Fun 𝐺 ↔ (Rel 𝐺 x dom 𝐺∃!z x𝐺z))
65simprbi 260 . . . . . . . 8 (Fun 𝐺x dom 𝐺∃!z x𝐺z)
74, 6anim12ci 322 . . . . . . 7 ((Fun 𝐹 Fun 𝐺) → (x dom 𝐺∃!z x𝐺z x dom 𝐺z∃*y z𝐹y))
8 r19.26 2435 . . . . . . 7 (x dom 𝐺(∃!z x𝐺z z∃*y z𝐹y) ↔ (x dom 𝐺∃!z x𝐺z x dom 𝐺z∃*y z𝐹y))
97, 8sylibr 137 . . . . . 6 ((Fun 𝐹 Fun 𝐺) → x dom 𝐺(∃!z x𝐺z z∃*y z𝐹y))
10 nfv 1418 . . . . . . . 8 y x𝐺z
1110euexex 1982 . . . . . . 7 ((∃!z x𝐺z z∃*y z𝐹y) → ∃*yz(x𝐺z z𝐹y))
1211ralimi 2378 . . . . . 6 (x dom 𝐺(∃!z x𝐺z z∃*y z𝐹y) → x dom 𝐺∃*yz(x𝐺z z𝐹y))
139, 12syl 14 . . . . 5 ((Fun 𝐹 Fun 𝐺) → x dom 𝐺∃*yz(x𝐺z z𝐹y))
14 ssralv 2998 . . . . 5 (dom (𝐹𝐺) ⊆ dom 𝐺 → (x dom 𝐺∃*yz(x𝐺z z𝐹y) → x dom (𝐹𝐺)∃*yz(x𝐺z z𝐹y)))
151, 13, 14mpsyl 59 . . . 4 ((Fun 𝐹 Fun 𝐺) → x dom (𝐹𝐺)∃*yz(x𝐺z z𝐹y))
16 df-br 3756 . . . . . . 7 (x(𝐹𝐺)y ↔ ⟨x, y (𝐹𝐺))
17 df-co 4297 . . . . . . . 8 (𝐹𝐺) = {⟨x, y⟩ ∣ z(x𝐺z z𝐹y)}
1817eleq2i 2101 . . . . . . 7 (⟨x, y (𝐹𝐺) ↔ ⟨x, y {⟨x, y⟩ ∣ z(x𝐺z z𝐹y)})
19 opabid 3985 . . . . . . 7 (⟨x, y {⟨x, y⟩ ∣ z(x𝐺z z𝐹y)} ↔ z(x𝐺z z𝐹y))
2016, 18, 193bitri 195 . . . . . 6 (x(𝐹𝐺)yz(x𝐺z z𝐹y))
2120mobii 1934 . . . . 5 (∃*y x(𝐹𝐺)y∃*yz(x𝐺z z𝐹y))
2221ralbii 2324 . . . 4 (x dom (𝐹𝐺)∃*y x(𝐹𝐺)yx dom (𝐹𝐺)∃*yz(x𝐺z z𝐹y))
2315, 22sylibr 137 . . 3 ((Fun 𝐹 Fun 𝐺) → x dom (𝐹𝐺)∃*y x(𝐹𝐺)y)
24 relco 4762 . . 3 Rel (𝐹𝐺)
2523, 24jctil 295 . 2 ((Fun 𝐹 Fun 𝐺) → (Rel (𝐹𝐺) x dom (𝐹𝐺)∃*y x(𝐹𝐺)y))
26 dffun7 4871 . 2 (Fun (𝐹𝐺) ↔ (Rel (𝐹𝐺) x dom (𝐹𝐺)∃*y x(𝐹𝐺)y))
2725, 26sylibr 137 1 ((Fun 𝐹 Fun 𝐺) → Fun (𝐹𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240  wex 1378   wcel 1390  ∃!weu 1897  ∃*wmo 1898  wral 2300  wss 2911  cop 3370   class class class wbr 3755  {copab 3808  dom cdm 4288  ccom 4292  Rel wrel 4293  Fun wfun 4839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847
This theorem is referenced by:  fnco  4950  f1co  5044  tposfun  5816
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