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Theorem fco 5107
Description: Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fco ((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)

Proof of Theorem fco
StepHypRef Expression
1 df-f 4956 . . 3 (𝐹:𝐵𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹𝐶))
2 df-f 4956 . . 3 (𝐺:𝐴𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺𝐵))
3 fnco 5058 . . . . . . 7 ((𝐹 Fn 𝐵𝐺 Fn 𝐴 ∧ ran 𝐺𝐵) → (𝐹𝐺) Fn 𝐴)
433expib 1142 . . . . . 6 (𝐹 Fn 𝐵 → ((𝐺 Fn 𝐴 ∧ ran 𝐺𝐵) → (𝐹𝐺) Fn 𝐴))
54adantr 270 . . . . 5 ((𝐹 Fn 𝐵 ∧ ran 𝐹𝐶) → ((𝐺 Fn 𝐴 ∧ ran 𝐺𝐵) → (𝐹𝐺) Fn 𝐴))
6 rncoss 4650 . . . . . . 7 ran (𝐹𝐺) ⊆ ran 𝐹
7 sstr 3016 . . . . . . 7 ((ran (𝐹𝐺) ⊆ ran 𝐹 ∧ ran 𝐹𝐶) → ran (𝐹𝐺) ⊆ 𝐶)
86, 7mpan 415 . . . . . 6 (ran 𝐹𝐶 → ran (𝐹𝐺) ⊆ 𝐶)
98adantl 271 . . . . 5 ((𝐹 Fn 𝐵 ∧ ran 𝐹𝐶) → ran (𝐹𝐺) ⊆ 𝐶)
105, 9jctird 310 . . . 4 ((𝐹 Fn 𝐵 ∧ ran 𝐹𝐶) → ((𝐺 Fn 𝐴 ∧ ran 𝐺𝐵) → ((𝐹𝐺) Fn 𝐴 ∧ ran (𝐹𝐺) ⊆ 𝐶)))
1110imp 122 . . 3 (((𝐹 Fn 𝐵 ∧ ran 𝐹𝐶) ∧ (𝐺 Fn 𝐴 ∧ ran 𝐺𝐵)) → ((𝐹𝐺) Fn 𝐴 ∧ ran (𝐹𝐺) ⊆ 𝐶))
121, 2, 11syl2anb 285 . 2 ((𝐹:𝐵𝐶𝐺:𝐴𝐵) → ((𝐹𝐺) Fn 𝐴 ∧ ran (𝐹𝐺) ⊆ 𝐶))
13 df-f 4956 . 2 ((𝐹𝐺):𝐴𝐶 ↔ ((𝐹𝐺) Fn 𝐴 ∧ ran (𝐹𝐺) ⊆ 𝐶))
1412, 13sylibr 132 1 ((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wss 2982  ran crn 4392  ccom 4395   Fn wfn 4947  wf 4948
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 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-fun 4954  df-fn 4955  df-f 4956
This theorem is referenced by:  fco2  5108  f1co  5152  foco  5167  ialgcvg  10637
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