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Theorem fcores 47017
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 6741 . . . . 5 (𝜑 → Fun 𝐹)
4 fcof 6760 . . . . 5 ((𝐺:𝐶𝐷 ∧ Fun 𝐹) → (𝐺𝐹):(𝐹𝐶)⟶𝐷)
51, 3, 4syl2anc 584 . . . 4 (𝜑 → (𝐺𝐹):(𝐹𝐶)⟶𝐷)
65ffnd 6738 . . 3 (𝜑 → (𝐺𝐹) Fn (𝐹𝐶))
7 fcores.p . . . 4 𝑃 = (𝐹𝐶)
87fneq2i 6667 . . 3 ((𝐺𝐹) Fn 𝑃 ↔ (𝐺𝐹) Fn (𝐹𝐶))
96, 8sylibr 234 . 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 47016 . 2 (𝜑 → (𝑌𝑋) Fn 𝑃)
1411fveq1i 6908 . . . . . 6 (𝑋𝑥) = ((𝐹𝑃)‘𝑥)
15 simpr 484 . . . . . . 7 ((𝜑𝑥𝑃) → 𝑥𝑃)
1615fvresd 6927 . . . . . 6 ((𝜑𝑥𝑃) → ((𝐹𝑃)‘𝑥) = (𝐹𝑥))
1714, 16eqtrid 2787 . . . . 5 ((𝜑𝑥𝑃) → (𝑋𝑥) = (𝐹𝑥))
1817fveq2d 6911 . . . 4 ((𝜑𝑥𝑃) → (𝑌‘(𝑋𝑥)) = (𝑌‘(𝐹𝑥)))
1912fveq1i 6908 . . . . 5 (𝑌‘(𝐹𝑥)) = ((𝐺𝐸)‘(𝐹𝑥))
20 cnvimass 6102 . . . . . . . . . . 11 (𝐹𝐶) ⊆ dom 𝐹
217, 20eqsstri 4030 . . . . . . . . . 10 𝑃 ⊆ dom 𝐹
2221sseli 3991 . . . . . . . . 9 (𝑥𝑃𝑥 ∈ dom 𝐹)
23 fvelrn 7096 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
243, 22, 23syl2an 596 . . . . . . . 8 ((𝜑𝑥𝑃) → (𝐹𝑥) ∈ ran 𝐹)
257eleq2i 2831 . . . . . . . . . 10 (𝑥𝑃𝑥 ∈ (𝐹𝐶))
2625biimpi 216 . . . . . . . . 9 (𝑥𝑃𝑥 ∈ (𝐹𝐶))
27 fvimacnvi 7072 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ (𝐹𝐶)) → (𝐹𝑥) ∈ 𝐶)
283, 26, 27syl2an 596 . . . . . . . 8 ((𝜑𝑥𝑃) → (𝐹𝑥) ∈ 𝐶)
2924, 28elind 4210 . . . . . . 7 ((𝜑𝑥𝑃) → (𝐹𝑥) ∈ (ran 𝐹𝐶))
3029, 10eleqtrrdi 2850 . . . . . 6 ((𝜑𝑥𝑃) → (𝐹𝑥) ∈ 𝐸)
3130fvresd 6927 . . . . 5 ((𝜑𝑥𝑃) → ((𝐺𝐸)‘(𝐹𝑥)) = (𝐺‘(𝐹𝑥)))
3219, 31eqtrid 2787 . . . 4 ((𝜑𝑥𝑃) → (𝑌‘(𝐹𝑥)) = (𝐺‘(𝐹𝑥)))
3318, 32eqtrd 2775 . . 3 ((𝜑𝑥𝑃) → (𝑌‘(𝑋𝑥)) = (𝐺‘(𝐹𝑥)))
342, 10, 7, 11fcoreslem3 47015 . . . . . 6 (𝜑𝑋:𝑃onto𝐸)
35 fof 6821 . . . . . 6 (𝑋:𝑃onto𝐸𝑋:𝑃𝐸)
3634, 35syl 17 . . . . 5 (𝜑𝑋:𝑃𝐸)
3736adantr 480 . . . 4 ((𝜑𝑥𝑃) → 𝑋:𝑃𝐸)
3837, 15fvco3d 7009 . . 3 ((𝜑𝑥𝑃) → ((𝑌𝑋)‘𝑥) = (𝑌‘(𝑋𝑥)))
392adantr 480 . . . 4 ((𝜑𝑥𝑃) → 𝐹:𝐴𝐵)
4021a1i 11 . . . . . 6 (𝜑𝑃 ⊆ dom 𝐹)
4140sselda 3995 . . . . 5 ((𝜑𝑥𝑃) → 𝑥 ∈ dom 𝐹)
422fdmd 6747 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐴)
4342eqcomd 2741 . . . . . . 7 (𝜑𝐴 = dom 𝐹)
4443eleq2d 2825 . . . . . 6 (𝜑 → (𝑥𝐴𝑥 ∈ dom 𝐹))
4544adantr 480 . . . . 5 ((𝜑𝑥𝑃) → (𝑥𝐴𝑥 ∈ dom 𝐹))
4641, 45mpbird 257 . . . 4 ((𝜑𝑥𝑃) → 𝑥𝐴)
4739, 46fvco3d 7009 . . 3 ((𝜑𝑥𝑃) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
4833, 38, 473eqtr4rd 2786 . 2 ((𝜑𝑥𝑃) → ((𝐺𝐹)‘𝑥) = ((𝑌𝑋)‘𝑥))
499, 13, 48eqfnfvd 7054 1 (𝜑 → (𝐺𝐹) = (𝑌𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cin 3962  wss 3963  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  ccom 5693  Fun wfun 6557   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571
This theorem is referenced by:  fcoresf1lem  47018  fcoresf1b  47020  fcoresfo  47021  fcoresfob  47022
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