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Theorem funco 4862
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 4524 . . . . 5 dom (𝐹𝐺) ⊆ dom 𝐺
2 funmo 4839 . . . . . . . . . 10 (Fun 𝐹∃*y z𝐹y)
32alrimiv 1732 . . . . . . . . 9 (Fun 𝐹z∃*y z𝐹y)
43ralrimivw 2367 . . . . . . . 8 (Fun 𝐹x dom 𝐺z∃*y z𝐹y)
5 dffun8 4851 . . . . . . . . 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 2415 . . . . . . 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 1398 . . . . . . . 8 y x𝐺z
1110euexex 1963 . . . . . . 7 ((∃!z x𝐺z z∃*y z𝐹y) → ∃*yz(x𝐺z z𝐹y))
1211ralimi 2358 . . . . . 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 2977 . . . . 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 3735 . . . . . . 7 (x(𝐹𝐺)y ↔ ⟨x, y (𝐹𝐺))
17 df-co 4277 . . . . . . . 8 (𝐹𝐺) = {⟨x, y⟩ ∣ z(x𝐺z z𝐹y)}
1817eleq2i 2082 . . . . . . 7 (⟨x, y (𝐹𝐺) ↔ ⟨x, y {⟨x, y⟩ ∣ z(x𝐺z z𝐹y)})
19 opabid 3964 . . . . . . 7 (⟨x, y {⟨x, y⟩ ∣ z(x𝐺z z𝐹y)} ↔ z(x𝐺z z𝐹y))
2016, 18, 193bitri 195 . . . . . 6 (x(𝐹𝐺)yz(x𝐺z z𝐹y))
2120mobii 1915 . . . . 5 (∃*y x(𝐹𝐺)y∃*yz(x𝐺z z𝐹y))
2221ralbii 2304 . . . 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 4742 . . 3 Rel (𝐹𝐺)
2523, 24jctil 295 . 2 ((Fun 𝐹 Fun 𝐺) → (Rel (𝐹𝐺) x dom (𝐹𝐺)∃*y x(𝐹𝐺)y))
26 dffun7 4850 . 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 1224  wex 1358   wcel 1370  ∃!weu 1878  ∃*wmo 1879  wral 2280  wss 2890  cop 3349   class class class wbr 3734  {copab 3787  dom cdm 4268  ccom 4272  Rel wrel 4273  Fun wfun 4819
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-fun 4827
This theorem is referenced by:  fnco  4929  f1co  5022  tposfun  5793
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