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Theorem fco 4999
Description: Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fco ((𝐹:B𝐶 𝐺:AB) → (𝐹𝐺):A𝐶)

Proof of Theorem fco
StepHypRef Expression
1 df-f 4849 . . 3 (𝐹:B𝐶 ↔ (𝐹 Fn B ran 𝐹𝐶))
2 df-f 4849 . . 3 (𝐺:AB ↔ (𝐺 Fn A ran 𝐺B))
3 fnco 4950 . . . . . . 7 ((𝐹 Fn B 𝐺 Fn A ran 𝐺B) → (𝐹𝐺) Fn A)
433expib 1106 . . . . . 6 (𝐹 Fn B → ((𝐺 Fn A ran 𝐺B) → (𝐹𝐺) Fn A))
54adantr 261 . . . . 5 ((𝐹 Fn B ran 𝐹𝐶) → ((𝐺 Fn A ran 𝐺B) → (𝐹𝐺) Fn A))
6 rncoss 4545 . . . . . . 7 ran (𝐹𝐺) ⊆ ran 𝐹
7 sstr 2947 . . . . . . 7 ((ran (𝐹𝐺) ⊆ ran 𝐹 ran 𝐹𝐶) → ran (𝐹𝐺) ⊆ 𝐶)
86, 7mpan 400 . . . . . 6 (ran 𝐹𝐶 → ran (𝐹𝐺) ⊆ 𝐶)
98adantl 262 . . . . 5 ((𝐹 Fn B ran 𝐹𝐶) → ran (𝐹𝐺) ⊆ 𝐶)
105, 9jctird 300 . . . 4 ((𝐹 Fn B ran 𝐹𝐶) → ((𝐺 Fn A ran 𝐺B) → ((𝐹𝐺) Fn A ran (𝐹𝐺) ⊆ 𝐶)))
1110imp 115 . . 3 (((𝐹 Fn B ran 𝐹𝐶) (𝐺 Fn A ran 𝐺B)) → ((𝐹𝐺) Fn A ran (𝐹𝐺) ⊆ 𝐶))
121, 2, 11syl2anb 275 . 2 ((𝐹:B𝐶 𝐺:AB) → ((𝐹𝐺) Fn A ran (𝐹𝐺) ⊆ 𝐶))
13 df-f 4849 . 2 ((𝐹𝐺):A𝐶 ↔ ((𝐹𝐺) Fn A ran (𝐹𝐺) ⊆ 𝐶))
1412, 13sylibr 137 1 ((𝐹:B𝐶 𝐺:AB) → (𝐹𝐺):A𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wss 2911  ran crn 4289  ccom 4292   Fn wfn 4840  wf 4841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849
This theorem is referenced by:  fco2  5000  f1co  5044  foco  5059
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