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Theorem fcoresfob 47022
Description: A composition is surjective iff the restriction of its first component to the minimum domain is surjective. (Contributed by GL and AV, 7-Oct-2024.)
Hypotheses
Ref Expression
fcores.f (𝜑𝐹:𝐴𝐵)
fcores.e 𝐸 = (ran 𝐹𝐶)
fcores.p 𝑃 = (𝐹𝐶)
fcores.x 𝑋 = (𝐹𝑃)
fcores.g (𝜑𝐺:𝐶𝐷)
fcores.y 𝑌 = (𝐺𝐸)
Assertion
Ref Expression
fcoresfob (𝜑 → ((𝐺𝐹):𝑃onto𝐷𝑌:𝐸onto𝐷))

Proof of Theorem fcoresfob
StepHypRef Expression
1 fcores.f . . . 4 (𝜑𝐹:𝐴𝐵)
21adantr 480 . . 3 ((𝜑 ∧ (𝐺𝐹):𝑃onto𝐷) → 𝐹:𝐴𝐵)
3 fcores.e . . 3 𝐸 = (ran 𝐹𝐶)
4 fcores.p . . 3 𝑃 = (𝐹𝐶)
5 fcores.x . . 3 𝑋 = (𝐹𝑃)
6 fcores.g . . . 4 (𝜑𝐺:𝐶𝐷)
76adantr 480 . . 3 ((𝜑 ∧ (𝐺𝐹):𝑃onto𝐷) → 𝐺:𝐶𝐷)
8 fcores.y . . 3 𝑌 = (𝐺𝐸)
9 simpr 484 . . 3 ((𝜑 ∧ (𝐺𝐹):𝑃onto𝐷) → (𝐺𝐹):𝑃onto𝐷)
102, 3, 4, 5, 7, 8, 9fcoresfo 47021 . 2 ((𝜑 ∧ (𝐺𝐹):𝑃onto𝐷) → 𝑌:𝐸onto𝐷)
111, 3, 4, 5fcoreslem3 47015 . . . . 5 (𝜑𝑋:𝑃onto𝐸)
1211anim1ci 616 . . . 4 ((𝜑𝑌:𝐸onto𝐷) → (𝑌:𝐸onto𝐷𝑋:𝑃onto𝐸))
13 foco 6835 . . . 4 ((𝑌:𝐸onto𝐷𝑋:𝑃onto𝐸) → (𝑌𝑋):𝑃onto𝐷)
1412, 13syl 17 . . 3 ((𝜑𝑌:𝐸onto𝐷) → (𝑌𝑋):𝑃onto𝐷)
151, 3, 4, 5, 6, 8fcores 47017 . . . . 5 (𝜑 → (𝐺𝐹) = (𝑌𝑋))
1615adantr 480 . . . 4 ((𝜑𝑌:𝐸onto𝐷) → (𝐺𝐹) = (𝑌𝑋))
17 foeq1 6817 . . . 4 ((𝐺𝐹) = (𝑌𝑋) → ((𝐺𝐹):𝑃onto𝐷 ↔ (𝑌𝑋):𝑃onto𝐷))
1816, 17syl 17 . . 3 ((𝜑𝑌:𝐸onto𝐷) → ((𝐺𝐹):𝑃onto𝐷 ↔ (𝑌𝑋):𝑃onto𝐷))
1914, 18mpbird 257 . 2 ((𝜑𝑌:𝐸onto𝐷) → (𝐺𝐹):𝑃onto𝐷)
2010, 19impbida 801 1 (𝜑 → ((𝐺𝐹):𝑃onto𝐷𝑌:𝐸onto𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  cin 3962  ccnv 5688  ran crn 5690  cres 5691  cima 5692  ccom 5693  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:  fcoresf1ob  47023
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