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Theorem fcoreslem4 45386
Description: Lemma 4 for fcores 45387. (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)
Assertion
Ref Expression
fcoreslem4 (𝜑 → (𝑌𝑋) Fn 𝑃)

Proof of Theorem fcoreslem4
StepHypRef Expression
1 fcores.g . . . . 5 (𝜑𝐺:𝐶𝐷)
21ffnd 6670 . . . 4 (𝜑𝐺 Fn 𝐶)
3 fcores.e . . . . . 6 𝐸 = (ran 𝐹𝐶)
43a1i 11 . . . . 5 (𝜑𝐸 = (ran 𝐹𝐶))
5 inss2 4190 . . . . 5 (ran 𝐹𝐶) ⊆ 𝐶
64, 5eqsstrdi 3999 . . . 4 (𝜑𝐸𝐶)
72, 6fnssresd 6626 . . 3 (𝜑 → (𝐺𝐸) Fn 𝐸)
8 fcores.y . . . 4 𝑌 = (𝐺𝐸)
98fneq1i 6600 . . 3 (𝑌 Fn 𝐸 ↔ (𝐺𝐸) Fn 𝐸)
107, 9sylibr 233 . 2 (𝜑𝑌 Fn 𝐸)
11 fcores.f . . . 4 (𝜑𝐹:𝐴𝐵)
12 fcores.p . . . 4 𝑃 = (𝐹𝐶)
13 fcores.x . . . 4 𝑋 = (𝐹𝑃)
1411, 3, 12, 13fcoreslem3 45385 . . 3 (𝜑𝑋:𝑃onto𝐸)
15 fofn 6759 . . 3 (𝑋:𝑃onto𝐸𝑋 Fn 𝑃)
1614, 15syl 17 . 2 (𝜑𝑋 Fn 𝑃)
1711, 3, 12, 13fcoreslem2 45384 . . 3 (𝜑 → ran 𝑋 = 𝐸)
18 eqimss 4001 . . 3 (ran 𝑋 = 𝐸 → ran 𝑋𝐸)
1917, 18syl 17 . 2 (𝜑 → ran 𝑋𝐸)
20 fnco 6619 . 2 ((𝑌 Fn 𝐸𝑋 Fn 𝑃 ∧ ran 𝑋𝐸) → (𝑌𝑋) Fn 𝑃)
2110, 16, 19, 20syl3anc 1372 1 (𝜑 → (𝑌𝑋) Fn 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cin 3910  wss 3911  ccnv 5633  ran crn 5635  cres 5636  cima 5637  ccom 5638   Fn wfn 6492  wf 6493  ontowfo 6495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503
This theorem is referenced by:  fcores  45387
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