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Theorem fnfocofob 46522
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 6081 . . . . 5 (𝐹 “ ran 𝐹) = dom 𝐹
2 fndm 6652 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
323ad2ant1 1130 . . . . 5 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → dom 𝐹 = 𝐴)
41, 3eqtr2id 2778 . . . 4 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (𝐹 “ ran 𝐹))
5 imaeq2 6054 . . . . 5 (ran 𝐹 = 𝐵 → (𝐹 “ ran 𝐹) = (𝐹𝐵))
653ad2ant3 1132 . . . 4 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → (𝐹 “ ran 𝐹) = (𝐹𝐵))
74, 6eqtrd 2765 . . 3 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (𝐹𝐵))
8 foeq2 6803 . . 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 6649 . . 3 (𝐹 Fn 𝐴 → Fun 𝐹)
11 id 22 . . 3 (𝐺:𝐵𝐶𝐺:𝐵𝐶)
12 eqimss2 4032 . . 3 (ran 𝐹 = 𝐵𝐵 ⊆ ran 𝐹)
13 funfocofob 46521 . . 3 ((Fun 𝐹𝐺:𝐵𝐶𝐵 ⊆ ran 𝐹) → ((𝐺𝐹):(𝐹𝐵)–onto𝐶𝐺:𝐵onto𝐶))
1410, 11, 12, 13syl3an 1157 . 2 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺𝐹):(𝐹𝐵)–onto𝐶𝐺:𝐵onto𝐶))
159, 14bitrd 278 1 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺𝐹):𝐴onto𝐶𝐺:𝐵onto𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wss 3939  ccnv 5671  dom cdm 5672  ran crn 5673  cima 5675  ccom 5676  Fun wfun 6537   Fn wfn 6538  wf 6539  ontowfo 6541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551
This theorem is referenced by:  focofob  46523  f1ocof1ob  46524
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