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Theorem funco 5021
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 4672 . . . . 5 dom (𝐹𝐺) ⊆ dom 𝐺
2 funmo 4998 . . . . . . . . . 10 (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
32alrimiv 1799 . . . . . . . . 9 (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
43ralrimivw 2443 . . . . . . . 8 (Fun 𝐹 → ∀𝑥 ∈ dom 𝐺𝑧∃*𝑦 𝑧𝐹𝑦)
5 dffun8 5010 . . . . . . . . 9 (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧))
65simprbi 269 . . . . . . . 8 (Fun 𝐺 → ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧)
74, 6anim12ci 332 . . . . . . 7 ((Fun 𝐹 ∧ Fun 𝐺) → (∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑥 ∈ dom 𝐺𝑧∃*𝑦 𝑧𝐹𝑦))
8 r19.26 2493 . . . . . . 7 (∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) ↔ (∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑥 ∈ dom 𝐺𝑧∃*𝑦 𝑧𝐹𝑦))
97, 8sylibr 132 . . . . . 6 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦))
10 nfv 1464 . . . . . . . 8 𝑦 𝑥𝐺𝑧
1110euexex 2030 . . . . . . 7 ((∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
1211ralimi 2434 . . . . . 6 (∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∀𝑥 ∈ dom 𝐺∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
139, 12syl 14 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom 𝐺∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
14 ssralv 3074 . . . . 5 (dom (𝐹𝐺) ⊆ dom 𝐺 → (∀𝑥 ∈ dom 𝐺∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦) → ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)))
151, 13, 14mpsyl 64 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
16 df-br 3823 . . . . . . 7 (𝑥(𝐹𝐺)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹𝐺))
17 df-co 4422 . . . . . . . 8 (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)}
1817eleq2i 2151 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐹𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)})
19 opabid 4060 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)} ↔ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2016, 18, 193bitri 204 . . . . . 6 (𝑥(𝐹𝐺)𝑦 ↔ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2120mobii 1982 . . . . 5 (∃*𝑦 𝑥(𝐹𝐺)𝑦 ↔ ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2221ralbii 2380 . . . 4 (∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦 𝑥(𝐹𝐺)𝑦 ↔ ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2315, 22sylibr 132 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦 𝑥(𝐹𝐺)𝑦)
24 relco 4897 . . 3 Rel (𝐹𝐺)
2523, 24jctil 305 . 2 ((Fun 𝐹 ∧ Fun 𝐺) → (Rel (𝐹𝐺) ∧ ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦 𝑥(𝐹𝐺)𝑦))
26 dffun7 5009 . 2 (Fun (𝐹𝐺) ↔ (Rel (𝐹𝐺) ∧ ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦 𝑥(𝐹𝐺)𝑦))
2725, 26sylibr 132 1 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1285  wex 1424  wcel 1436  ∃!weu 1945  ∃*wmo 1946  wral 2355  wss 2988  cop 3434   class class class wbr 3822  {copab 3875  dom cdm 4413  ccom 4417  Rel wrel 4418  Fun wfun 4977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-fun 4985
This theorem is referenced by:  fnco  5089  f1co  5193  tposfun  5981  casefun  6723  caseinj  6727  djufun  6731  djuinj  6733
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