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Theorem fnfocofob 45318
Description: If the domain of a function 𝐺 equals the range of a function 𝐹, then the composition (𝐺𝐹) is surjective iff 𝐺 is surjective. (Contributed by GL and AV, 29-Sep-2024.)
Assertion
Ref Expression
fnfocofob ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺𝐹):𝐴onto𝐶𝐺:𝐵onto𝐶))

Proof of Theorem fnfocofob
StepHypRef Expression
1 cnvimarndm 6035 . . . . 5 (𝐹 “ ran 𝐹) = dom 𝐹
2 fndm 6606 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
323ad2ant1 1134 . . . . 5 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → dom 𝐹 = 𝐴)
41, 3eqtr2id 2790 . . . 4 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (𝐹 “ ran 𝐹))
5 imaeq2 6010 . . . . 5 (ran 𝐹 = 𝐵 → (𝐹 “ ran 𝐹) = (𝐹𝐵))
653ad2ant3 1136 . . . 4 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → (𝐹 “ ran 𝐹) = (𝐹𝐵))
74, 6eqtrd 2777 . . 3 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (𝐹𝐵))
8 foeq2 6754 . . 3 (𝐴 = (𝐹𝐵) → ((𝐺𝐹):𝐴onto𝐶 ↔ (𝐺𝐹):(𝐹𝐵)–onto𝐶))
97, 8syl 17 . 2 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺𝐹):𝐴onto𝐶 ↔ (𝐺𝐹):(𝐹𝐵)–onto𝐶))
10 fnfun 6603 . . 3 (𝐹 Fn 𝐴 → Fun 𝐹)
11 id 22 . . 3 (𝐺:𝐵𝐶𝐺:𝐵𝐶)
12 eqimss2 4002 . . 3 (ran 𝐹 = 𝐵𝐵 ⊆ ran 𝐹)
13 funfocofob 45317 . . 3 ((Fun 𝐹𝐺:𝐵𝐶𝐵 ⊆ ran 𝐹) → ((𝐺𝐹):(𝐹𝐵)–onto𝐶𝐺:𝐵onto𝐶))
1410, 11, 12, 13syl3an 1161 . 2 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺𝐹):(𝐹𝐵)–onto𝐶𝐺:𝐵onto𝐶))
159, 14bitrd 279 1 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺𝐹):𝐴onto𝐶𝐺:𝐵onto𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1088   = wceq 1542  wss 3911  ccnv 5633  dom cdm 5634  ran crn 5635  cima 5637  ccom 5638  Fun wfun 6491   Fn wfn 6492  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-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  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-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  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-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505
This theorem is referenced by:  focofob  45319  f1ocof1ob  45320
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