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Theorem foco 6835
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 6822 . . . 4 (𝐺:𝐴onto𝐵 → Fun 𝐺)
32adantl 481 . . 3 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → Fun 𝐺)
4 forn 6824 . . . . 5 (𝐺:𝐴onto𝐵 → ran 𝐺 = 𝐵)
5 eqimss2 4055 . . . . 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 6834 . . 3 ((𝐹:𝐵onto𝐶 ∧ Fun 𝐺𝐵 ⊆ ran 𝐺) → (𝐹𝐺):(𝐺𝐵)–onto𝐶)
91, 3, 7, 8syl3anc 1370 . 2 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):(𝐺𝐵)–onto𝐶)
10 focnvimacdmdm 6833 . . . . 5 (𝐺:𝐴onto𝐵 → (𝐺𝐵) = 𝐴)
1110eqcomd 2741 . . . 4 (𝐺:𝐴onto𝐵𝐴 = (𝐺𝐵))
1211adantl 481 . . 3 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → 𝐴 = (𝐺𝐵))
13 foeq2 6818 . . 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 1537  wss 3963  ccnv 5688  ran crn 5690  cima 5692  ccom 5693  Fun wfun 6557  ontowfo 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569
This theorem is referenced by:  f1oco  6872  wdomtr  9613  fin1a2lem7  10444  cofull  17988  sursubmefmnd  18922  uniiccdif  25627  fcoresfob  47022
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