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Theorem focofob 47030
Description: If the domain of a function 𝐺 equals the range of a function 𝐹, then the composition (𝐺𝐹) is surjective iff 𝐺 and 𝐹 as function to the domain of 𝐺 are both surjective. Symmetric version of fnfocofob 47029 including the fact that 𝐹 is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024.) (Proof shortened by AV, 29-Sep-2024.)
Assertion
Ref Expression
focofob ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴onto𝐷 ↔ (𝐹:𝐴onto𝐶𝐺:𝐶onto𝐷)))

Proof of Theorem focofob
StepHypRef Expression
1 ffn 6737 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fnfocofob 47029 . . 3 ((𝐹 Fn 𝐴𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴onto𝐷𝐺:𝐶onto𝐷))
31, 2syl3an1 1162 . 2 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴onto𝐷𝐺:𝐶onto𝐷))
4 dffn4 6827 . . . . . 6 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
51, 4sylib 218 . . . . 5 (𝐹:𝐴𝐵𝐹:𝐴onto→ran 𝐹)
653ad2ant1 1132 . . . 4 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴onto→ran 𝐹)
7 foeq3 6819 . . . . 5 (ran 𝐹 = 𝐶 → (𝐹:𝐴onto→ran 𝐹𝐹:𝐴onto𝐶))
873ad2ant3 1134 . . . 4 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴onto→ran 𝐹𝐹:𝐴onto𝐶))
96, 8mpbid 232 . . 3 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴onto𝐶)
109biantrurd 532 . 2 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐺:𝐶onto𝐷 ↔ (𝐹:𝐴onto𝐶𝐺:𝐶onto𝐷)))
113, 10bitrd 279 1 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴onto𝐷 ↔ (𝐹:𝐴onto𝐶𝐺:𝐶onto𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  ran crn 5690  ccom 5693   Fn wfn 6558  wf 6559  ontowfo 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571
This theorem is referenced by: (None)
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