Theorem fcores 47659

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 6700	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
4		fcof 6719	. . . . 5 ⊢ ((𝐺:𝐶⟶𝐷 ∧ Fun 𝐹) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)⟶𝐷)
5	1, 3, 4	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)⟶𝐷)
6	5	ffnd 6696	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) Fn (◡𝐹 “ 𝐶))
7		fcores.p	. . . 4 ⊢ 𝑃 = (◡𝐹 “ 𝐶)
8	7	fneq2i 6623	. . 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 47658	. 2 ⊢ (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃)
14	11	fveq1i 6872	. . . . . 6 ⊢ (𝑋‘𝑥) = ((𝐹 ↾ 𝑃)‘𝑥)
15		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ 𝑃)
16	15	fvresd 6891	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐹 ↾ 𝑃)‘𝑥) = (𝐹‘𝑥))
17	14, 16	eqtrid 2812	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑋‘𝑥) = (𝐹‘𝑥))
18	17	fveq2d 6875	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑌‘(𝑋‘𝑥)) = (𝑌‘(𝐹‘𝑥)))
19	12	fveq1i 6872	. . . . 5 ⊢ (𝑌‘(𝐹‘𝑥)) = ((𝐺 ↾ 𝐸)‘(𝐹‘𝑥))
20		cnvimass 6075	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝐶) ⊆ dom 𝐹
21	7, 20	eqsstri 3985	. . . . . . . . . 10 ⊢ 𝑃 ⊆ dom 𝐹
22	21	sseli 3935	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ dom 𝐹)
23		fvelrn 7061	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
24	3, 22, 23	syl2an 607	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ ran 𝐹)
25	7	eleq2i 2857	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑃 ↔ 𝑥 ∈ (◡𝐹 “ 𝐶))
26	25	biimpi 219	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ (◡𝐹 “ 𝐶))
27		fvimacnvi 7037	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (◡𝐹 “ 𝐶)) → (𝐹‘𝑥) ∈ 𝐶)
28	3, 26, 27	syl2an 607	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ 𝐶)
29	24, 28	elind 4155	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ (ran 𝐹 ∩ 𝐶))
30	29, 10	eleqtrrdi 2876	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ 𝐸)
31	30	fvresd 6891	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐺 ↾ 𝐸)‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
32	19, 31	eqtrid 2812	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑌‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
33	18, 32	eqtrd 2800	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑌‘(𝑋‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
34	2, 10, 7, 11	fcoreslem3 47657	. . . . . 6 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
35		fof 6782	. . . . . 6 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋:𝑃⟶𝐸)
36	34, 35	syl 18	. . . . 5 ⊢ (𝜑 → 𝑋:𝑃⟶𝐸)
37	36	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑋:𝑃⟶𝐸)
38	37, 15	fvco3d 6972	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝑌 ∘ 𝑋)‘𝑥) = (𝑌‘(𝑋‘𝑥)))
39	2	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝐹:𝐴⟶𝐵)
40	21	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑃 ⊆ dom 𝐹)
41	40	sselda 3939	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ dom 𝐹)
42	2	fdmd 6706	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
43	42	eqcomd 2771	. . . . . . 7 ⊢ (𝜑 → 𝐴 = dom 𝐹)
44	43	eleq2d 2851	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ dom 𝐹))
45	44	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ dom 𝐹))
46	41, 45	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ 𝐴)
47	39, 46	fvco3d 6972	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
48	33, 38, 47	3eqtr4rd 2811	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑥) = ((𝑌 ∘ 𝑋)‘𝑥))
49	9, 13, 48	eqfnfvd 7018	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ∩ cin 3906   ⊆ wss 3907  ^◡ccnv 5651  dom cdm 5652  ran crn 5653   ↾ cres 5654   “ cima 5655   ∘ ccom 5656  Fun wfun 6519   Fn wfn 6520  ⟶wf 6521  –onto→wfo 6523  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fo 6531  df-fv 6533

This theorem is referenced by:  fcoresf1lem  47660  fcoresf1b  47662  fcoresfo  47663  fcoresfob  47664