ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  funco GIF version

Theorem funco 4967
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 4628 . . . . 5 dom (𝐹𝐺) ⊆ dom 𝐺
2 funmo 4944 . . . . . . . . . 10 (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
32alrimiv 1770 . . . . . . . . 9 (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
43ralrimivw 2410 . . . . . . . 8 (Fun 𝐹 → ∀𝑥 ∈ dom 𝐺𝑧∃*𝑦 𝑧𝐹𝑦)
5 dffun8 4956 . . . . . . . . 9 (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧))
65simprbi 264 . . . . . . . 8 (Fun 𝐺 → ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧)
74, 6anim12ci 326 . . . . . . 7 ((Fun 𝐹 ∧ Fun 𝐺) → (∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑥 ∈ dom 𝐺𝑧∃*𝑦 𝑧𝐹𝑦))
8 r19.26 2458 . . . . . . 7 (∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) ↔ (∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑥 ∈ dom 𝐺𝑧∃*𝑦 𝑧𝐹𝑦))
97, 8sylibr 141 . . . . . 6 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦))
10 nfv 1437 . . . . . . . 8 𝑦 𝑥𝐺𝑧
1110euexex 2001 . . . . . . 7 ((∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
1211ralimi 2401 . . . . . 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 3031 . . . . 5 (dom (𝐹𝐺) ⊆ dom 𝐺 → (∀𝑥 ∈ dom 𝐺∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦) → ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)))
151, 13, 14mpsyl 63 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
16 df-br 3792 . . . . . . 7 (𝑥(𝐹𝐺)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹𝐺))
17 df-co 4381 . . . . . . . 8 (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)}
1817eleq2i 2120 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐹𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)})
19 opabid 4021 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)} ↔ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2016, 18, 193bitri 199 . . . . . 6 (𝑥(𝐹𝐺)𝑦 ↔ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2120mobii 1953 . . . . 5 (∃*𝑦 𝑥(𝐹𝐺)𝑦 ↔ ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2221ralbii 2347 . . . 4 (∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦 𝑥(𝐹𝐺)𝑦 ↔ ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
2315, 22sylibr 141 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦 𝑥(𝐹𝐺)𝑦)
24 relco 4846 . . 3 Rel (𝐹𝐺)
2523, 24jctil 299 . 2 ((Fun 𝐹 ∧ Fun 𝐺) → (Rel (𝐹𝐺) ∧ ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦 𝑥(𝐹𝐺)𝑦))
26 dffun7 4955 . 2 (Fun (𝐹𝐺) ↔ (Rel (𝐹𝐺) ∧ ∀𝑥 ∈ dom (𝐹𝐺)∃*𝑦 𝑥(𝐹𝐺)𝑦))
2725, 26sylibr 141 1 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257  wex 1397  wcel 1409  ∃!weu 1916  ∃*wmo 1917  wral 2323  wss 2944  cop 3405   class class class wbr 3791  {copab 3844  dom cdm 4372  ccom 4376  Rel wrel 4377  Fun wfun 4923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-fun 4931
This theorem is referenced by:  fnco  5034  f1co  5128  tposfun  5905
  Copyright terms: Public domain W3C validator