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Theorem foco 6771
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 484 . . 3 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → 𝐹:𝐵onto𝐶)
2 fofun 6758 . . . 4 (𝐺:𝐴onto𝐵 → Fun 𝐺)
32adantl 483 . . 3 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → Fun 𝐺)
4 forn 6760 . . . . 5 (𝐺:𝐴onto𝐵 → ran 𝐺 = 𝐵)
5 eqimss2 4002 . . . . 5 (ran 𝐺 = 𝐵𝐵 ⊆ ran 𝐺)
64, 5syl 17 . . . 4 (𝐺:𝐴onto𝐵𝐵 ⊆ ran 𝐺)
76adantl 483 . . 3 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → 𝐵 ⊆ ran 𝐺)
8 focofo 6770 . . 3 ((𝐹:𝐵onto𝐶 ∧ Fun 𝐺𝐵 ⊆ ran 𝐺) → (𝐹𝐺):(𝐺𝐵)–onto𝐶)
91, 3, 7, 8syl3anc 1372 . 2 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):(𝐺𝐵)–onto𝐶)
10 focnvimacdmdm 6769 . . . . 5 (𝐺:𝐴onto𝐵 → (𝐺𝐵) = 𝐴)
1110eqcomd 2743 . . . 4 (𝐺:𝐴onto𝐵𝐴 = (𝐺𝐵))
1211adantl 483 . . 3 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → 𝐴 = (𝐺𝐵))
13 foeq2 6754 . . 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 397   = wceq 1542  wss 3911  ccnv 5633  ran crn 5635  cima 5637  ccom 5638  Fun wfun 6491  ontowfo 6495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503
This theorem is referenced by:  f1oco  6808  wdomtr  9512  fin1a2lem7  10343  cofull  17822  sursubmefmnd  18707  uniiccdif  24945  fcoresfob  45313
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