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Theorem fnfocofob 46359
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 6075 . . . . 5 (𝐹 “ ran 𝐹) = dom 𝐹
2 fndm 6646 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
323ad2ant1 1130 . . . . 5 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → dom 𝐹 = 𝐴)
41, 3eqtr2id 2779 . . . 4 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (𝐹 “ ran 𝐹))
5 imaeq2 6049 . . . . 5 (ran 𝐹 = 𝐵 → (𝐹 “ ran 𝐹) = (𝐹𝐵))
653ad2ant3 1132 . . . 4 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → (𝐹 “ ran 𝐹) = (𝐹𝐵))
74, 6eqtrd 2766 . . 3 ((𝐹 Fn 𝐴𝐺:𝐵𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (𝐹𝐵))
8 foeq2 6796 . . 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 6643 . . 3 (𝐹 Fn 𝐴 → Fun 𝐹)
11 id 22 . . 3 (𝐺:𝐵𝐶𝐺:𝐵𝐶)
12 eqimss2 4036 . . 3 (ran 𝐹 = 𝐵𝐵 ⊆ ran 𝐹)
13 funfocofob 46358 . . 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 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 1084   = wceq 1533  wss 3943  ccnv 5668  dom cdm 5669  ran crn 5670  cima 5672  ccom 5673  Fun wfun 6531   Fn wfn 6532  wf 6533  ontowfo 6535
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545
This theorem is referenced by:  focofob  46360  f1ocof1ob  46361
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