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Theorem fnco 5038
Description: Composition of two functions. (Contributed by NM, 22-May-2006.)
Assertion
Ref Expression
fnco ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)

Proof of Theorem fnco
StepHypRef Expression
1 fnfun 5027 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fnfun 5027 . . . 4 (𝐺 Fn 𝐵 → Fun 𝐺)
3 funco 4970 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2an 283 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → Fun (𝐹𝐺))
543adant3 959 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → Fun (𝐹𝐺))
6 fndm 5029 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76sseq2d 3028 . . . . . 6 (𝐹 Fn 𝐴 → (ran 𝐺 ⊆ dom 𝐹 ↔ ran 𝐺𝐴))
87biimpar 291 . . . . 5 ((𝐹 Fn 𝐴 ∧ ran 𝐺𝐴) → ran 𝐺 ⊆ dom 𝐹)
9 dmcosseq 4631 . . . . 5 (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹𝐺) = dom 𝐺)
108, 9syl 14 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = dom 𝐺)
11103adant2 958 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = dom 𝐺)
12 fndm 5029 . . . 4 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
13123ad2ant2 961 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom 𝐺 = 𝐵)
1411, 13eqtrd 2114 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = 𝐵)
15 df-fn 4935 . 2 ((𝐹𝐺) Fn 𝐵 ↔ (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = 𝐵))
165, 14, 15sylanbrc 408 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wss 2974  dom cdm 4371  ran crn 4372  ccom 4375  Fun wfun 4926   Fn wfn 4927
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-fun 4934  df-fn 4935
This theorem is referenced by:  fco  5087  fnfco  5096
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