Theorem fcores 48106

Description: Every composite function (𝐺 ∘ 𝐹) 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	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fcores.e	⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
fcores.p	⊢ 𝑃 = (◡𝐹 “ 𝐶)
fcores.x	⊢ 𝑋 = (𝐹 ↾ 𝑃)
fcores.g	⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
fcores.y	⊢ 𝑌 = (𝐺 ↾ 𝐸)

Assertion

Ref	Expression
fcores	⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))

Proof of Theorem fcores

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fcores.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
2		fcores.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3	2	ffund 6712	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
4		fcof 6731	. . . . 5 ⊢ ((𝐺:𝐶⟶𝐷 ∧ Fun 𝐹) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)⟶𝐷)
5	1, 3, 4	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)⟶𝐷)
6	5	ffnd 6708	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) Fn (◡𝐹 “ 𝐶))
7		fcores.p	. . . 4 ⊢ 𝑃 = (◡𝐹 “ 𝐶)
8	7	fneq2i 6635	. . 3 ⊢ ((𝐺 ∘ 𝐹) Fn 𝑃 ↔ (𝐺 ∘ 𝐹) Fn (◡𝐹 “ 𝐶))
9	6, 8	sylibr 237	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) Fn 𝑃)
10		fcores.e	. . 3 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
11		fcores.x	. . 3 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
12		fcores.y	. . 3 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
13	2, 10, 7, 11, 1, 12	fcoreslem4 48105	. 2 ⊢ (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃)
14	11	fveq1i 6884	. . . . . 6 ⊢ (𝑋‘𝑥) = ((𝐹 ↾ 𝑃)‘𝑥)
15		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ 𝑃)
16	15	fvresd 6903	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐹 ↾ 𝑃)‘𝑥) = (𝐹‘𝑥))
17	14, 16	eqtrid 2808	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑋‘𝑥) = (𝐹‘𝑥))
18	17	fveq2d 6887	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑌‘(𝑋‘𝑥)) = (𝑌‘(𝐹‘𝑥)))
19	12	fveq1i 6884	. . . . 5 ⊢ (𝑌‘(𝐹‘𝑥)) = ((𝐺 ↾ 𝐸)‘(𝐹‘𝑥))
20		cnvimass 6197	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝐶) ⊆ dom 𝐹
21	7, 20	eqsstri 3977	. . . . . . . . . 10 ⊢ 𝑃 ⊆ dom 𝐹
22	21	sseli 3927	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ dom 𝐹)
23		fvelrn 7074	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
24	3, 22, 23	syl2an 608	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ ran 𝐹)
25	7	eleq2i 2853	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑃 ↔ 𝑥 ∈ (◡𝐹 “ 𝐶))
26	25	biimpi 219	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ (◡𝐹 “ 𝐶))
27		fvimacnvi 7049	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (◡𝐹 “ 𝐶)) → (𝐹‘𝑥) ∈ 𝐶)
28	3, 26, 27	syl2an 608	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ 𝐶)
29	24, 28	elind 4146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ (ran 𝐹 ∩ 𝐶))
30	29, 10	eleqtrrdi 2872	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ 𝐸)
31	30	fvresd 6903	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐺 ↾ 𝐸)‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
32	19, 31	eqtrid 2808	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑌‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
33	18, 32	eqtrd 2796	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑌‘(𝑋‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
34	2, 10, 7, 11	fcoreslem3 48104	. . . . . 6 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
35		fof 6794	. . . . . 6 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋:𝑃⟶𝐸)
36	34, 35	syl 18	. . . . 5 ⊢ (𝜑 → 𝑋:𝑃⟶𝐸)
37	36	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑋:𝑃⟶𝐸)
38	37, 15	fvco3d 6984	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝑌 ∘ 𝑋)‘𝑥) = (𝑌‘(𝑋‘𝑥)))
39	2	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝐹:𝐴⟶𝐵)
40	21	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑃 ⊆ dom 𝐹)
41	40	sselda 3931	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ dom 𝐹)
42	2	fdmd 6718	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
43	42	eqcomd 2767	. . . . . . 7 ⊢ (𝜑 → 𝐴 = dom 𝐹)
44	43	eleq2d 2847	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ dom 𝐹))
45	44	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ dom 𝐹))
46	41, 45	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ 𝐴)
47	39, 46	fvco3d 6984	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
48	33, 38, 47	3eqtr4rd 2807	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑥) = ((𝑌 ∘ 𝑋)‘𝑥))
49	9, 13, 48	eqfnfvd 7030	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∩ cin 3898   ⊆ wss 3899  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654   ∘ ccom 5655  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545

This theorem is used by:  fcoresf1lem  48107  fcoresf1b  48109  fcoresfo  48110  fcoresfob  48111