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Theorem focofo 6834
Description: Composition of onto functions. Generalisation of foco 6835. (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 6821 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2 fcof 6760 . . . 4 ((𝐹:𝐴𝐵 ∧ Fun 𝐺) → (𝐹𝐺):(𝐺𝐴)⟶𝐵)
31, 2sylan 580 . . 3 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺) → (𝐹𝐺):(𝐺𝐴)⟶𝐵)
433adant3 1131 . 2 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → (𝐹𝐺):(𝐺𝐴)⟶𝐵)
5 rnco 6274 . . 3 ran (𝐹𝐺) = ran (𝐹 ↾ ran 𝐺)
61freld 6743 . . . . . 6 (𝐹:𝐴onto𝐵 → Rel 𝐹)
763ad2ant1 1132 . . . . 5 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → Rel 𝐹)
8 fdm 6746 . . . . . . . . 9 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
98eqcomd 2741 . . . . . . . 8 (𝐹:𝐴𝐵𝐴 = dom 𝐹)
101, 9syl 17 . . . . . . 7 (𝐹:𝐴onto𝐵𝐴 = dom 𝐹)
1110sseq1d 4027 . . . . . 6 (𝐹:𝐴onto𝐵 → (𝐴 ⊆ ran 𝐺 ↔ dom 𝐹 ⊆ ran 𝐺))
1211biimpa 476 . . . . 5 ((𝐹:𝐴onto𝐵𝐴 ⊆ ran 𝐺) → dom 𝐹 ⊆ ran 𝐺)
13 relssres 6042 . . . . . 6 ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → (𝐹 ↾ ran 𝐺) = 𝐹)
1413rneqd 5952 . . . . 5 ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = ran 𝐹)
157, 12, 143imp3i2an 1344 . . . 4 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = ran 𝐹)
16 forn 6824 . . . . 5 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
17163ad2ant1 1132 . . . 4 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → ran 𝐹 = 𝐵)
1815, 17eqtrd 2775 . . 3 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = 𝐵)
195, 18eqtrid 2787 . 2 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → ran (𝐹𝐺) = 𝐵)
20 dffo2 6825 . 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 1086   = wceq 1537  wss 3963  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  ccom 5693  Rel wrel 5694  Fun wfun 6557  wf 6559  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:  foco  6835  funfocofob  47028
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