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Theorem fco 5381
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 5220 . . 3 (𝐹:𝐵𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹𝐶))
2 df-f 5220 . . 3 (𝐺:𝐴𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺𝐵))
3 fnco 5324 . . . . . . 7 ((𝐹 Fn 𝐵𝐺 Fn 𝐴 ∧ ran 𝐺𝐵) → (𝐹𝐺) Fn 𝐴)
433expib 1206 . . . . . 6 (𝐹 Fn 𝐵 → ((𝐺 Fn 𝐴 ∧ ran 𝐺𝐵) → (𝐹𝐺) Fn 𝐴))
54adantr 276 . . . . 5 ((𝐹 Fn 𝐵 ∧ ran 𝐹𝐶) → ((𝐺 Fn 𝐴 ∧ ran 𝐺𝐵) → (𝐹𝐺) Fn 𝐴))
6 rncoss 4897 . . . . . . 7 ran (𝐹𝐺) ⊆ ran 𝐹
7 sstr 3163 . . . . . . 7 ((ran (𝐹𝐺) ⊆ ran 𝐹 ∧ ran 𝐹𝐶) → ran (𝐹𝐺) ⊆ 𝐶)
86, 7mpan 424 . . . . . 6 (ran 𝐹𝐶 → ran (𝐹𝐺) ⊆ 𝐶)
98adantl 277 . . . . 5 ((𝐹 Fn 𝐵 ∧ ran 𝐹𝐶) → ran (𝐹𝐺) ⊆ 𝐶)
105, 9jctird 317 . . . 4 ((𝐹 Fn 𝐵 ∧ ran 𝐹𝐶) → ((𝐺 Fn 𝐴 ∧ ran 𝐺𝐵) → ((𝐹𝐺) Fn 𝐴 ∧ ran (𝐹𝐺) ⊆ 𝐶)))
1110imp 124 . . 3 (((𝐹 Fn 𝐵 ∧ ran 𝐹𝐶) ∧ (𝐺 Fn 𝐴 ∧ ran 𝐺𝐵)) → ((𝐹𝐺) Fn 𝐴 ∧ ran (𝐹𝐺) ⊆ 𝐶))
121, 2, 11syl2anb 291 . 2 ((𝐹:𝐵𝐶𝐺:𝐴𝐵) → ((𝐹𝐺) Fn 𝐴 ∧ ran (𝐹𝐺) ⊆ 𝐶))
13 df-f 5220 . 2 ((𝐹𝐺):𝐴𝐶 ↔ ((𝐹𝐺) Fn 𝐴 ∧ ran (𝐹𝐺) ⊆ 𝐶))
1412, 13sylibr 134 1 ((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wss 3129  ran crn 4627  ccom 4630   Fn wfn 5211  wf 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-fun 5218  df-fn 5219  df-f 5220
This theorem is referenced by:  fco2  5382  f1co  5433  foco  5448  mapen  6845  ctm  7107  enomnilem  7135  enmkvlem  7158  enwomnilem  7166  fnn0nninf  10436  fsumcl2lem  11405  fsumadd  11413  fprodmul  11598  algcvg  12047  mhmco  12873  cnco  13691  cnptopco  13692  lmtopcnp  13720  cnmpt11  13753  cnmpt21  13761  comet  13969  cnmet  14000  cncfco  14048  limccnpcntop  14114  dvcoapbr  14141  dvcjbr  14142  dvcj  14143
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