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Theorem fcoreslem4 46983
Description: Lemma 4 for fcores 46984. (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 6750 . . . 4 (𝜑𝐺 Fn 𝐶)
3 fcores.e . . . . . 6 𝐸 = (ran 𝐹𝐶)
43a1i 11 . . . . 5 (𝜑𝐸 = (ran 𝐹𝐶))
5 inss2 4259 . . . . 5 (ran 𝐹𝐶) ⊆ 𝐶
64, 5eqsstrdi 4063 . . . 4 (𝜑𝐸𝐶)
72, 6fnssresd 6706 . . 3 (𝜑 → (𝐺𝐸) Fn 𝐸)
8 fcores.y . . . 4 𝑌 = (𝐺𝐸)
98fneq1i 6678 . . 3 (𝑌 Fn 𝐸 ↔ (𝐺𝐸) Fn 𝐸)
107, 9sylibr 234 . 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 46982 . . 3 (𝜑𝑋:𝑃onto𝐸)
15 fofn 6838 . . 3 (𝑋:𝑃onto𝐸𝑋 Fn 𝑃)
1614, 15syl 17 . 2 (𝜑𝑋 Fn 𝑃)
1711, 3, 12, 13fcoreslem2 46981 . . 3 (𝜑 → ran 𝑋 = 𝐸)
18 eqimss 4067 . . 3 (ran 𝑋 = 𝐸 → ran 𝑋𝐸)
1917, 18syl 17 . 2 (𝜑 → ran 𝑋𝐸)
20 fnco 6699 . 2 ((𝑌 Fn 𝐸𝑋 Fn 𝑃 ∧ ran 𝑋𝐸) → (𝑌𝑋) Fn 𝑃)
2110, 16, 19, 20syl3anc 1371 1 (𝜑 → (𝑌𝑋) Fn 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cin 3975  wss 3976  ccnv 5699  ran crn 5701  cres 5702  cima 5703  ccom 5704   Fn wfn 6570  wf 6571  ontowfo 6573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-fun 6577  df-fn 6578  df-f 6579  df-fo 6581
This theorem is referenced by:  fcores  46984
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