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Theorem focofo 6770
Description: Composition of onto functions. Generalisation of foco 6771. (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 6757 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2 fcof 6692 . . . 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 6205 . . 3 ran (𝐹𝐺) = ran (𝐹 ↾ ran 𝐺)
61freld 6675 . . . . . 6 (𝐹:𝐴onto𝐵 → Rel 𝐹)
763ad2ant1 1134 . . . . 5 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → Rel 𝐹)
8 fdm 6678 . . . . . . . . 9 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
98eqcomd 2743 . . . . . . . 8 (𝐹:𝐴𝐵𝐴 = dom 𝐹)
101, 9syl 17 . . . . . . 7 (𝐹:𝐴onto𝐵𝐴 = dom 𝐹)
1110sseq1d 3976 . . . . . 6 (𝐹:𝐴onto𝐵 → (𝐴 ⊆ ran 𝐺 ↔ dom 𝐹 ⊆ ran 𝐺))
1211biimpa 478 . . . . 5 ((𝐹:𝐴onto𝐵𝐴 ⊆ ran 𝐺) → dom 𝐹 ⊆ ran 𝐺)
13 relssres 5979 . . . . . 6 ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → (𝐹 ↾ ran 𝐺) = 𝐹)
1413rneqd 5894 . . . . 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 6760 . . . . 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 6761 . 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 3911  ccnv 5633  dom cdm 5634  ran crn 5635  cres 5636  cima 5637  ccom 5638  Rel wrel 5639  Fun wfun 6491  wf 6493  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:  foco  6771  funfocofob  45317
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