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Theorem foco 6834
Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.) (Proof shortened by AV, 29-Sep-2024.)
Assertion
Ref Expression
foco ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):𝐴onto𝐶)

Proof of Theorem foco
StepHypRef Expression
1 simpl 482 . . 3 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → 𝐹:𝐵onto𝐶)
2 fofun 6821 . . . 4 (𝐺:𝐴onto𝐵 → Fun 𝐺)
32adantl 481 . . 3 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → Fun 𝐺)
4 forn 6823 . . . . 5 (𝐺:𝐴onto𝐵 → ran 𝐺 = 𝐵)
5 eqimss2 4043 . . . . 5 (ran 𝐺 = 𝐵𝐵 ⊆ ran 𝐺)
64, 5syl 17 . . . 4 (𝐺:𝐴onto𝐵𝐵 ⊆ ran 𝐺)
76adantl 481 . . 3 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → 𝐵 ⊆ ran 𝐺)
8 focofo 6833 . . 3 ((𝐹:𝐵onto𝐶 ∧ Fun 𝐺𝐵 ⊆ ran 𝐺) → (𝐹𝐺):(𝐺𝐵)–onto𝐶)
91, 3, 7, 8syl3anc 1373 . 2 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):(𝐺𝐵)–onto𝐶)
10 focnvimacdmdm 6832 . . . . 5 (𝐺:𝐴onto𝐵 → (𝐺𝐵) = 𝐴)
1110eqcomd 2743 . . . 4 (𝐺:𝐴onto𝐵𝐴 = (𝐺𝐵))
1211adantl 481 . . 3 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → 𝐴 = (𝐺𝐵))
13 foeq2 6817 . . 3 (𝐴 = (𝐺𝐵) → ((𝐹𝐺):𝐴onto𝐶 ↔ (𝐹𝐺):(𝐺𝐵)–onto𝐶))
1412, 13syl 17 . 2 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → ((𝐹𝐺):𝐴onto𝐶 ↔ (𝐹𝐺):(𝐺𝐵)–onto𝐶))
159, 14mpbird 257 1 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):𝐴onto𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wss 3951  ccnv 5684  ran crn 5686  cima 5688  ccom 5689  Fun wfun 6555  ontowfo 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567
This theorem is referenced by:  f1oco  6871  wdomtr  9615  fin1a2lem7  10446  cofull  17981  sursubmefmnd  18909  uniiccdif  25613  fcoresfob  47084
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