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Theorem fcoresfo 47088
Description: If a composition is surjective, then the restriction of its first component to the minimum domain is surjective. (Contributed by AV, 17-Sep-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)
fcoresfo.s (𝜑 → (𝐺𝐹):𝑃onto𝐷)
Assertion
Ref Expression
fcoresfo (𝜑𝑌:𝐸onto𝐷)

Proof of Theorem fcoresfo
StepHypRef Expression
1 fcores.g . . . 4 (𝜑𝐺:𝐶𝐷)
2 fcores.e . . . . . 6 𝐸 = (ran 𝐹𝐶)
32a1i 11 . . . . 5 (𝜑𝐸 = (ran 𝐹𝐶))
4 inss2 4237 . . . . 5 (ran 𝐹𝐶) ⊆ 𝐶
53, 4eqsstrdi 4027 . . . 4 (𝜑𝐸𝐶)
61, 5fssresd 6774 . . 3 (𝜑 → (𝐺𝐸):𝐸𝐷)
7 fcores.y . . . 4 𝑌 = (𝐺𝐸)
87feq1i 6726 . . 3 (𝑌:𝐸𝐷 ↔ (𝐺𝐸):𝐸𝐷)
96, 8sylibr 234 . 2 (𝜑𝑌:𝐸𝐷)
10 fcores.f . . . 4 (𝜑𝐹:𝐴𝐵)
11 fcores.p . . . 4 𝑃 = (𝐹𝐶)
12 fcores.x . . . 4 𝑋 = (𝐹𝑃)
1310, 2, 11, 12fcoreslem3 47082 . . 3 (𝜑𝑋:𝑃onto𝐸)
14 fof 6819 . . 3 (𝑋:𝑃onto𝐸𝑋:𝑃𝐸)
1513, 14syl 17 . 2 (𝜑𝑋:𝑃𝐸)
16 fcoresfo.s . . 3 (𝜑 → (𝐺𝐹):𝑃onto𝐷)
1710, 2, 11, 12, 1, 7fcores 47084 . . . . 5 (𝜑 → (𝐺𝐹) = (𝑌𝑋))
1817eqcomd 2742 . . . 4 (𝜑 → (𝑌𝑋) = (𝐺𝐹))
19 foeq1 6815 . . . 4 ((𝑌𝑋) = (𝐺𝐹) → ((𝑌𝑋):𝑃onto𝐷 ↔ (𝐺𝐹):𝑃onto𝐷))
2018, 19syl 17 . . 3 (𝜑 → ((𝑌𝑋):𝑃onto𝐷 ↔ (𝐺𝐹):𝑃onto𝐷))
2116, 20mpbird 257 . 2 (𝜑 → (𝑌𝑋):𝑃onto𝐷)
22 foco2 7128 . 2 ((𝑌:𝐸𝐷𝑋:𝑃𝐸 ∧ (𝑌𝑋):𝑃onto𝐷) → 𝑌:𝐸onto𝐷)
239, 15, 21, 22syl3anc 1372 1 (𝜑𝑌:𝐸onto𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  cin 3949  ccnv 5683  ran crn 5685  cres 5686  cima 5687  ccom 5688  wf 6556  ontowfo 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568
This theorem is referenced by:  fcoresfob  47089  funfocofob  47095
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