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Theorem fcoreslem4 46510
Description: Lemma 4 for fcores 46511. (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 6717 . . . 4 (𝜑𝐺 Fn 𝐶)
3 fcores.e . . . . . 6 𝐸 = (ran 𝐹𝐶)
43a1i 11 . . . . 5 (𝜑𝐸 = (ran 𝐹𝐶))
5 inss2 4224 . . . . 5 (ran 𝐹𝐶) ⊆ 𝐶
64, 5eqsstrdi 4027 . . . 4 (𝜑𝐸𝐶)
72, 6fnssresd 6673 . . 3 (𝜑 → (𝐺𝐸) Fn 𝐸)
8 fcores.y . . . 4 𝑌 = (𝐺𝐸)
98fneq1i 6645 . . 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 46509 . . 3 (𝜑𝑋:𝑃onto𝐸)
15 fofn 6807 . . 3 (𝑋:𝑃onto𝐸𝑋 Fn 𝑃)
1614, 15syl 17 . 2 (𝜑𝑋 Fn 𝑃)
1711, 3, 12, 13fcoreslem2 46508 . . 3 (𝜑 → ran 𝑋 = 𝐸)
18 eqimss 4031 . . 3 (ran 𝑋 = 𝐸 → ran 𝑋𝐸)
1917, 18syl 17 . 2 (𝜑 → ran 𝑋𝐸)
20 fnco 6666 . 2 ((𝑌 Fn 𝐸𝑋 Fn 𝑃 ∧ ran 𝑋𝐸) → (𝑌𝑋) Fn 𝑃)
2110, 16, 19, 20syl3anc 1368 1 (𝜑 → (𝑌𝑋) Fn 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cin 3939  wss 3940  ccnv 5671  ran crn 5673  cres 5674  cima 5675  ccom 5676   Fn wfn 6537  wf 6538  ontowfo 6540
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 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548
This theorem is referenced by:  fcores  46511
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