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Theorem funco 5131
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 4776 . . . . 5 dom (𝐹𝐺) ⊆ dom 𝐺
2 funmo 5106 . . . . . . . . . 10 (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
32alrimiv 1828 . . . . . . . . 9 (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
43ralrimivw 2481 . . . . . . . 8 (Fun 𝐹 → ∀𝑥 ∈ dom 𝐺𝑧∃*𝑦 𝑧𝐹𝑦)
5 dffun8 5119 . . . . . . . . 9 (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧))
65simprbi 271 . . . . . . . 8 (Fun 𝐺 → ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧)
74, 6anim12ci 335 . . . . . . 7 ((Fun 𝐹 ∧ Fun 𝐺) → (∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑥 ∈ dom 𝐺𝑧∃*𝑦 𝑧𝐹𝑦))
8 r19.26 2533 . . . . . . 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 1491 . . . . . . . 8 𝑦 𝑥𝐺𝑧
1110euexex 2060 . . . . . . 7 ((∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
1211ralimi 2470 . . . . . 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 3129 . . . . 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 3898 . . . . . . 7 (𝑥(𝐹𝐺)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹𝐺))
17 df-co 4516 . . . . . . . 8 (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)}
1817eleq2i 2182 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐹𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)})
19 opabid 4147 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)} ↔ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2016, 18, 193bitri 205 . . . . . 6 (𝑥(𝐹𝐺)𝑦 ↔ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2120mobii 2012 . . . . 5 (∃*𝑦 𝑥(𝐹𝐺)𝑦 ↔ ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2221ralbii 2416 . . . 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 5005 . . 3 Rel (𝐹𝐺)
2523, 24jctil 308 . 2 ((Fun 𝐹 ∧ Fun 𝐺) → (Rel (𝐹𝐺) ∧ ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦 𝑥(𝐹𝐺)𝑦))
26 dffun7 5118 . 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 1312  wex 1451  wcel 1463  ∃!weu 1975  ∃*wmo 1976  wral 2391  wss 3039  cop 3498   class class class wbr 3897  {copab 3956  dom cdm 4507  ccom 4511  Rel wrel 4512  Fun wfun 5085
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-fun 5093
This theorem is referenced by:  fnco  5199  f1co  5308  tposfun  6123  casefun  6936  caseinj  6940  caseinl  6942  caseinr  6943  djufun  6955  djuinj  6957  ctssdccl  6962
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