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Theorem fcores 45387
Description: Every composite function (𝐺𝐹) can be written as composition of restrictions of the composed functions (to their minimum domains). (Contributed by GL and 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
fcores (𝜑 → (𝐺𝐹) = (𝑌𝑋))

Proof of Theorem fcores
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fcores.g . . . . 5 (𝜑𝐺:𝐶𝐷)
2 fcores.f . . . . . 6 (𝜑𝐹:𝐴𝐵)
32ffund 6673 . . . . 5 (𝜑 → Fun 𝐹)
4 fcof 6692 . . . . 5 ((𝐺:𝐶𝐷 ∧ Fun 𝐹) → (𝐺𝐹):(𝐹𝐶)⟶𝐷)
51, 3, 4syl2anc 585 . . . 4 (𝜑 → (𝐺𝐹):(𝐹𝐶)⟶𝐷)
65ffnd 6670 . . 3 (𝜑 → (𝐺𝐹) Fn (𝐹𝐶))
7 fcores.p . . . 4 𝑃 = (𝐹𝐶)
87fneq2i 6601 . . 3 ((𝐺𝐹) Fn 𝑃 ↔ (𝐺𝐹) Fn (𝐹𝐶))
96, 8sylibr 233 . 2 (𝜑 → (𝐺𝐹) Fn 𝑃)
10 fcores.e . . 3 𝐸 = (ran 𝐹𝐶)
11 fcores.x . . 3 𝑋 = (𝐹𝑃)
12 fcores.y . . 3 𝑌 = (𝐺𝐸)
132, 10, 7, 11, 1, 12fcoreslem4 45386 . 2 (𝜑 → (𝑌𝑋) Fn 𝑃)
1411fveq1i 6844 . . . . . 6 (𝑋𝑥) = ((𝐹𝑃)‘𝑥)
15 simpr 486 . . . . . . 7 ((𝜑𝑥𝑃) → 𝑥𝑃)
1615fvresd 6863 . . . . . 6 ((𝜑𝑥𝑃) → ((𝐹𝑃)‘𝑥) = (𝐹𝑥))
1714, 16eqtrid 2785 . . . . 5 ((𝜑𝑥𝑃) → (𝑋𝑥) = (𝐹𝑥))
1817fveq2d 6847 . . . 4 ((𝜑𝑥𝑃) → (𝑌‘(𝑋𝑥)) = (𝑌‘(𝐹𝑥)))
1912fveq1i 6844 . . . . 5 (𝑌‘(𝐹𝑥)) = ((𝐺𝐸)‘(𝐹𝑥))
20 cnvimass 6034 . . . . . . . . . . 11 (𝐹𝐶) ⊆ dom 𝐹
217, 20eqsstri 3979 . . . . . . . . . 10 𝑃 ⊆ dom 𝐹
2221sseli 3941 . . . . . . . . 9 (𝑥𝑃𝑥 ∈ dom 𝐹)
23 fvelrn 7028 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
243, 22, 23syl2an 597 . . . . . . . 8 ((𝜑𝑥𝑃) → (𝐹𝑥) ∈ ran 𝐹)
257eleq2i 2826 . . . . . . . . . 10 (𝑥𝑃𝑥 ∈ (𝐹𝐶))
2625biimpi 215 . . . . . . . . 9 (𝑥𝑃𝑥 ∈ (𝐹𝐶))
27 fvimacnvi 7003 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ (𝐹𝐶)) → (𝐹𝑥) ∈ 𝐶)
283, 26, 27syl2an 597 . . . . . . . 8 ((𝜑𝑥𝑃) → (𝐹𝑥) ∈ 𝐶)
2924, 28elind 4155 . . . . . . 7 ((𝜑𝑥𝑃) → (𝐹𝑥) ∈ (ran 𝐹𝐶))
3029, 10eleqtrrdi 2845 . . . . . 6 ((𝜑𝑥𝑃) → (𝐹𝑥) ∈ 𝐸)
3130fvresd 6863 . . . . 5 ((𝜑𝑥𝑃) → ((𝐺𝐸)‘(𝐹𝑥)) = (𝐺‘(𝐹𝑥)))
3219, 31eqtrid 2785 . . . 4 ((𝜑𝑥𝑃) → (𝑌‘(𝐹𝑥)) = (𝐺‘(𝐹𝑥)))
3318, 32eqtrd 2773 . . 3 ((𝜑𝑥𝑃) → (𝑌‘(𝑋𝑥)) = (𝐺‘(𝐹𝑥)))
342, 10, 7, 11fcoreslem3 45385 . . . . . 6 (𝜑𝑋:𝑃onto𝐸)
35 fof 6757 . . . . . 6 (𝑋:𝑃onto𝐸𝑋:𝑃𝐸)
3634, 35syl 17 . . . . 5 (𝜑𝑋:𝑃𝐸)
3736adantr 482 . . . 4 ((𝜑𝑥𝑃) → 𝑋:𝑃𝐸)
3837, 15fvco3d 6942 . . 3 ((𝜑𝑥𝑃) → ((𝑌𝑋)‘𝑥) = (𝑌‘(𝑋𝑥)))
392adantr 482 . . . 4 ((𝜑𝑥𝑃) → 𝐹:𝐴𝐵)
4021a1i 11 . . . . . 6 (𝜑𝑃 ⊆ dom 𝐹)
4140sselda 3945 . . . . 5 ((𝜑𝑥𝑃) → 𝑥 ∈ dom 𝐹)
422fdmd 6680 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐴)
4342eqcomd 2739 . . . . . . 7 (𝜑𝐴 = dom 𝐹)
4443eleq2d 2820 . . . . . 6 (𝜑 → (𝑥𝐴𝑥 ∈ dom 𝐹))
4544adantr 482 . . . . 5 ((𝜑𝑥𝑃) → (𝑥𝐴𝑥 ∈ dom 𝐹))
4641, 45mpbird 257 . . . 4 ((𝜑𝑥𝑃) → 𝑥𝐴)
4739, 46fvco3d 6942 . . 3 ((𝜑𝑥𝑃) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
4833, 38, 473eqtr4rd 2784 . 2 ((𝜑𝑥𝑃) → ((𝐺𝐹)‘𝑥) = ((𝑌𝑋)‘𝑥))
499, 13, 48eqfnfvd 6986 1 (𝜑 → (𝐺𝐹) = (𝑌𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cin 3910  wss 3911  ccnv 5633  dom cdm 5634  ran crn 5635  cres 5636  cima 5637  ccom 5638  Fun wfun 6491   Fn wfn 6492  wf 6493  ontowfo 6495  cfv 6497
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-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  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-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  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-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505
This theorem is referenced by:  fcoresf1lem  45388  fcoresf1b  45390  fcoresfo  45391  fcoresfob  45392
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