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Theorem fcoresfo 46482
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 4232 . . . . 5 (ran 𝐹𝐶) ⊆ 𝐶
53, 4eqsstrdi 4036 . . . 4 (𝜑𝐸𝐶)
61, 5fssresd 6769 . . 3 (𝜑 → (𝐺𝐸):𝐸𝐷)
7 fcores.y . . . 4 𝑌 = (𝐺𝐸)
87feq1i 6718 . . 3 (𝑌:𝐸𝐷 ↔ (𝐺𝐸):𝐸𝐷)
96, 8sylibr 233 . 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 46476 . . 3 (𝜑𝑋:𝑃onto𝐸)
14 fof 6816 . . 3 (𝑋:𝑃onto𝐸𝑋:𝑃𝐸)
1513, 14syl 17 . 2 (𝜑𝑋:𝑃𝐸)
16 fcoresfo.s . . 3 (𝜑 → (𝐺𝐹):𝑃onto𝐷)
1710, 2, 11, 12, 1, 7fcores 46478 . . . . 5 (𝜑 → (𝐺𝐹) = (𝑌𝑋))
1817eqcomd 2734 . . . 4 (𝜑 → (𝑌𝑋) = (𝐺𝐹))
19 foeq1 6812 . . . 4 ((𝑌𝑋) = (𝐺𝐹) → ((𝑌𝑋):𝑃onto𝐷 ↔ (𝐺𝐹):𝑃onto𝐷))
2018, 19syl 17 . . 3 (𝜑 → ((𝑌𝑋):𝑃onto𝐷 ↔ (𝐺𝐹):𝑃onto𝐷))
2116, 20mpbird 256 . 2 (𝜑 → (𝑌𝑋):𝑃onto𝐷)
22 foco2 7124 . 2 ((𝑌:𝐸𝐷𝑋:𝑃𝐸 ∧ (𝑌𝑋):𝑃onto𝐷) → 𝑌:𝐸onto𝐷)
239, 15, 21, 22syl3anc 1368 1 (𝜑𝑌:𝐸onto𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  cin 3948  ccnv 5681  ran crn 5683  cres 5684  cima 5685  ccom 5686  wf 6549  ontowfo 6551
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561
This theorem is referenced by:  fcoresfob  46483  funfocofob  46487
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