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Theorem foco 5116
 Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
foco ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):𝐴onto𝐶)

Proof of Theorem foco
StepHypRef Expression
1 dffo2 5110 . . 3 (𝐹:𝐵onto𝐶 ↔ (𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶))
2 dffo2 5110 . . 3 (𝐺:𝐴onto𝐵 ↔ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵))
3 fco 5056 . . . . 5 ((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
43ad2ant2r 478 . . . 4 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → (𝐹𝐺):𝐴𝐶)
5 fdm 5050 . . . . . . . 8 (𝐹:𝐵𝐶 → dom 𝐹 = 𝐵)
6 eqtr3 2059 . . . . . . . 8 ((dom 𝐹 = 𝐵 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
75, 6sylan 267 . . . . . . 7 ((𝐹:𝐵𝐶 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
8 rncoeq 4605 . . . . . . . . 9 (dom 𝐹 = ran 𝐺 → ran (𝐹𝐺) = ran 𝐹)
98eqeq1d 2048 . . . . . . . 8 (dom 𝐹 = ran 𝐺 → (ran (𝐹𝐺) = 𝐶 ↔ ran 𝐹 = 𝐶))
109biimpar 281 . . . . . . 7 ((dom 𝐹 = ran 𝐺 ∧ ran 𝐹 = 𝐶) → ran (𝐹𝐺) = 𝐶)
117, 10sylan 267 . . . . . 6 (((𝐹:𝐵𝐶 ∧ ran 𝐺 = 𝐵) ∧ ran 𝐹 = 𝐶) → ran (𝐹𝐺) = 𝐶)
1211an32s 502 . . . . 5 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ ran 𝐺 = 𝐵) → ran (𝐹𝐺) = 𝐶)
1312adantrl 447 . . . 4 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → ran (𝐹𝐺) = 𝐶)
144, 13jca 290 . . 3 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
151, 2, 14syl2anb 275 . 2 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
16 dffo2 5110 . 2 ((𝐹𝐺):𝐴onto𝐶 ↔ ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
1715, 16sylibr 137 1 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):𝐴onto𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  dom cdm 4345  ran crn 4346   ∘ ccom 4349  ⟶wf 4898  –onto→wfo 4900 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908 This theorem is referenced by:  f1oco  5149
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