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Theorem funco 5172
 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 4817 . . . . 5 dom (𝐹𝐺) ⊆ dom 𝐺
2 funmo 5147 . . . . . . . . . 10 (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
32alrimiv 1847 . . . . . . . . 9 (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
43ralrimivw 2510 . . . . . . . 8 (Fun 𝐹 → ∀𝑥 ∈ dom 𝐺𝑧∃*𝑦 𝑧𝐹𝑦)
5 dffun8 5160 . . . . . . . . 9 (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧))
65simprbi 273 . . . . . . . 8 (Fun 𝐺 → ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧)
74, 6anim12ci 337 . . . . . . 7 ((Fun 𝐹 ∧ Fun 𝐺) → (∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑥 ∈ dom 𝐺𝑧∃*𝑦 𝑧𝐹𝑦))
8 r19.26 2562 . . . . . . 7 (∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) ↔ (∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑥 ∈ dom 𝐺𝑧∃*𝑦 𝑧𝐹𝑦))
97, 8sylibr 133 . . . . . 6 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦))
10 nfv 1509 . . . . . . . 8 𝑦 𝑥𝐺𝑧
1110euexex 2085 . . . . . . 7 ((∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
1211ralimi 2499 . . . . . 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 3167 . . . . 5 (dom (𝐹𝐺) ⊆ dom 𝐺 → (∀𝑥 ∈ dom 𝐺∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦) → ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)))
151, 13, 14mpsyl 65 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
16 df-br 3939 . . . . . . 7 (𝑥(𝐹𝐺)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹𝐺))
17 df-co 4557 . . . . . . . 8 (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)}
1817eleq2i 2207 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐹𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)})
19 opabid 4188 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)} ↔ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2016, 18, 193bitri 205 . . . . . 6 (𝑥(𝐹𝐺)𝑦 ↔ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2120mobii 2037 . . . . 5 (∃*𝑦 𝑥(𝐹𝐺)𝑦 ↔ ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2221ralbii 2445 . . . 4 (∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦 𝑥(𝐹𝐺)𝑦 ↔ ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2315, 22sylibr 133 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦 𝑥(𝐹𝐺)𝑦)
24 relco 5046 . . 3 Rel (𝐹𝐺)
2523, 24jctil 310 . 2 ((Fun 𝐹 ∧ Fun 𝐺) → (Rel (𝐹𝐺) ∧ ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦 𝑥(𝐹𝐺)𝑦))
26 dffun7 5159 . 2 (Fun (𝐹𝐺) ↔ (Rel (𝐹𝐺) ∧ ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦 𝑥(𝐹𝐺)𝑦))
2725, 26sylibr 133 1 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330  ∃wex 1469   ∈ wcel 1481  ∃!weu 2000  ∃*wmo 2001  ∀wral 2417   ⊆ wss 3077  ⟨cop 3536   class class class wbr 3938  {copab 3997  dom cdm 4548   ∘ ccom 4552  Rel wrel 4553  Fun wfun 5126 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-br 3939  df-opab 3999  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-fun 5134 This theorem is referenced by:  fnco  5240  f1co  5349  tposfun  6166  casefun  6980  caseinj  6984  caseinl  6986  caseinr  6987  djufun  6999  djuinj  7001  ctssdccl  7006
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