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Theorem focofo 6833
Description: Composition of onto functions. Generalisation of foco 6834. (Contributed by AV, 29-Sep-2024.)
Assertion
Ref Expression
focofo ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → (𝐹𝐺):(𝐺𝐴)–onto𝐵)

Proof of Theorem focofo
StepHypRef Expression
1 fof 6820 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2 fcof 6759 . . . 4 ((𝐹:𝐴𝐵 ∧ Fun 𝐺) → (𝐹𝐺):(𝐺𝐴)⟶𝐵)
31, 2sylan 580 . . 3 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺) → (𝐹𝐺):(𝐺𝐴)⟶𝐵)
433adant3 1133 . 2 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → (𝐹𝐺):(𝐺𝐴)⟶𝐵)
5 rnco 6272 . . 3 ran (𝐹𝐺) = ran (𝐹 ↾ ran 𝐺)
61freld 6742 . . . . . 6 (𝐹:𝐴onto𝐵 → Rel 𝐹)
763ad2ant1 1134 . . . . 5 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → Rel 𝐹)
8 fdm 6745 . . . . . . . . 9 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
98eqcomd 2743 . . . . . . . 8 (𝐹:𝐴𝐵𝐴 = dom 𝐹)
101, 9syl 17 . . . . . . 7 (𝐹:𝐴onto𝐵𝐴 = dom 𝐹)
1110sseq1d 4015 . . . . . 6 (𝐹:𝐴onto𝐵 → (𝐴 ⊆ ran 𝐺 ↔ dom 𝐹 ⊆ ran 𝐺))
1211biimpa 476 . . . . 5 ((𝐹:𝐴onto𝐵𝐴 ⊆ ran 𝐺) → dom 𝐹 ⊆ ran 𝐺)
13 relssres 6040 . . . . . 6 ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → (𝐹 ↾ ran 𝐺) = 𝐹)
1413rneqd 5949 . . . . 5 ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = ran 𝐹)
157, 12, 143imp3i2an 1346 . . . 4 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = ran 𝐹)
16 forn 6823 . . . . 5 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
17163ad2ant1 1134 . . . 4 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → ran 𝐹 = 𝐵)
1815, 17eqtrd 2777 . . 3 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = 𝐵)
195, 18eqtrid 2789 . 2 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → ran (𝐹𝐺) = 𝐵)
20 dffo2 6824 . 2 ((𝐹𝐺):(𝐺𝐴)–onto𝐵 ↔ ((𝐹𝐺):(𝐺𝐴)⟶𝐵 ∧ ran (𝐹𝐺) = 𝐵))
214, 19, 20sylanbrc 583 1 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → (𝐹𝐺):(𝐺𝐴)–onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wss 3951  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  ccom 5689  Rel wrel 5690  Fun wfun 6555  wf 6557  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:  foco  6834  funfocofob  47090
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