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Theorem fnco 5108
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 5097 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fnfun 5097 . . . 4 (𝐺 Fn 𝐵 → Fun 𝐺)
3 funco 5040 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2an 283 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → Fun (𝐹𝐺))
543adant3 963 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → Fun (𝐹𝐺))
6 fndm 5099 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76sseq2d 3052 . . . . . 6 (𝐹 Fn 𝐴 → (ran 𝐺 ⊆ dom 𝐹 ↔ ran 𝐺𝐴))
87biimpar 291 . . . . 5 ((𝐹 Fn 𝐴 ∧ ran 𝐺𝐴) → ran 𝐺 ⊆ dom 𝐹)
9 dmcosseq 4692 . . . . 5 (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹𝐺) = dom 𝐺)
108, 9syl 14 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = dom 𝐺)
11103adant2 962 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = dom 𝐺)
12 fndm 5099 . . . 4 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
13123ad2ant2 965 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom 𝐺 = 𝐵)
1411, 13eqtrd 2120 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = 𝐵)
15 df-fn 5005 . 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 924   = wceq 1289  wss 2997  dom cdm 4428  ran crn 4429  ccom 4432  Fun wfun 4996   Fn wfn 4997
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-fun 5004  df-fn 5005
This theorem is referenced by:  fco  5161  fnfco  5170  updjudhcoinlf  6750  updjudhcoinrg  6751
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