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Theorem foco 6810
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 6797 . . . 4 (𝐺:𝐴onto𝐵 → Fun 𝐺)
32adantl 481 . . 3 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → Fun 𝐺)
4 forn 6799 . . . . 5 (𝐺:𝐴onto𝐵 → ran 𝐺 = 𝐵)
5 eqimss2 4034 . . . . 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 6809 . . 3 ((𝐹:𝐵onto𝐶 ∧ Fun 𝐺𝐵 ⊆ ran 𝐺) → (𝐹𝐺):(𝐺𝐵)–onto𝐶)
91, 3, 7, 8syl3anc 1368 . 2 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):(𝐺𝐵)–onto𝐶)
10 focnvimacdmdm 6808 . . . . 5 (𝐺:𝐴onto𝐵 → (𝐺𝐵) = 𝐴)
1110eqcomd 2730 . . . 4 (𝐺:𝐴onto𝐵𝐴 = (𝐺𝐵))
1211adantl 481 . . 3 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → 𝐴 = (𝐺𝐵))
13 foeq2 6793 . . 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 205  wa 395   = wceq 1533  wss 3941  ccnv 5666  ran crn 5668  cima 5670  ccom 5671  Fun wfun 6528  ontowfo 6532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-fun 6536  df-fn 6537  df-f 6538  df-fo 6540
This theorem is referenced by:  f1oco  6847  wdomtr  9567  fin1a2lem7  10398  cofull  17892  sursubmefmnd  18817  uniiccdif  25451  fcoresfob  46327
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