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Theorem focofo 6819
Description: Composition of onto functions. Generalisation of foco 6820. (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 6806 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2 fcof 6741 . . . 4 ((𝐹:𝐴𝐵 ∧ Fun 𝐺) → (𝐹𝐺):(𝐺𝐴)⟶𝐵)
31, 2sylan 581 . . 3 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺) → (𝐹𝐺):(𝐺𝐴)⟶𝐵)
433adant3 1133 . 2 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → (𝐹𝐺):(𝐺𝐴)⟶𝐵)
5 rnco 6252 . . 3 ran (𝐹𝐺) = ran (𝐹 ↾ ran 𝐺)
61freld 6724 . . . . . 6 (𝐹:𝐴onto𝐵 → Rel 𝐹)
763ad2ant1 1134 . . . . 5 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → Rel 𝐹)
8 fdm 6727 . . . . . . . . 9 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
98eqcomd 2739 . . . . . . . 8 (𝐹:𝐴𝐵𝐴 = dom 𝐹)
101, 9syl 17 . . . . . . 7 (𝐹:𝐴onto𝐵𝐴 = dom 𝐹)
1110sseq1d 4014 . . . . . 6 (𝐹:𝐴onto𝐵 → (𝐴 ⊆ ran 𝐺 ↔ dom 𝐹 ⊆ ran 𝐺))
1211biimpa 478 . . . . 5 ((𝐹:𝐴onto𝐵𝐴 ⊆ ran 𝐺) → dom 𝐹 ⊆ ran 𝐺)
13 relssres 6023 . . . . . 6 ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → (𝐹 ↾ ran 𝐺) = 𝐹)
1413rneqd 5938 . . . . 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 6809 . . . . 5 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
17163ad2ant1 1134 . . . 4 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → ran 𝐹 = 𝐵)
1815, 17eqtrd 2773 . . 3 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = 𝐵)
195, 18eqtrid 2785 . 2 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → ran (𝐹𝐺) = 𝐵)
20 dffo2 6810 . 2 ((𝐹𝐺):(𝐺𝐴)–onto𝐵 ↔ ((𝐹𝐺):(𝐺𝐴)⟶𝐵 ∧ ran (𝐹𝐺) = 𝐵))
214, 19, 20sylanbrc 584 1 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → (𝐹𝐺):(𝐺𝐴)–onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wss 3949  ccnv 5676  dom cdm 5677  ran crn 5678  cres 5679  cima 5680  ccom 5681  Rel wrel 5682  Fun wfun 6538  wf 6540  ontowfo 6542
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550
This theorem is referenced by:  foco  6820  funfocofob  45786
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