Theorem fcores 47068

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 6692	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
4		fcof 6711	. . . . 5 ⊢ ((𝐺:𝐶⟶𝐷 ∧ Fun 𝐹) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)⟶𝐷)
5	1, 3, 4	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)⟶𝐷)
6	5	ffnd 6689	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) Fn (◡𝐹 “ 𝐶))
7		fcores.p	. . . 4 ⊢ 𝑃 = (◡𝐹 “ 𝐶)
8	7	fneq2i 6616	. . 3 ⊢ ((𝐺 ∘ 𝐹) Fn 𝑃 ↔ (𝐺 ∘ 𝐹) Fn (◡𝐹 “ 𝐶))
9	6, 8	sylibr 234	. 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 47067	. 2 ⊢ (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃)
14	11	fveq1i 6859	. . . . . 6 ⊢ (𝑋‘𝑥) = ((𝐹 ↾ 𝑃)‘𝑥)
15		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ 𝑃)
16	15	fvresd 6878	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐹 ↾ 𝑃)‘𝑥) = (𝐹‘𝑥))
17	14, 16	eqtrid 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑋‘𝑥) = (𝐹‘𝑥))
18	17	fveq2d 6862	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑌‘(𝑋‘𝑥)) = (𝑌‘(𝐹‘𝑥)))
19	12	fveq1i 6859	. . . . 5 ⊢ (𝑌‘(𝐹‘𝑥)) = ((𝐺 ↾ 𝐸)‘(𝐹‘𝑥))
20		cnvimass 6053	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝐶) ⊆ dom 𝐹
21	7, 20	eqsstri 3993	. . . . . . . . . 10 ⊢ 𝑃 ⊆ dom 𝐹
22	21	sseli 3942	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ dom 𝐹)
23		fvelrn 7048	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
24	3, 22, 23	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ ran 𝐹)
25	7	eleq2i 2820	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑃 ↔ 𝑥 ∈ (◡𝐹 “ 𝐶))
26	25	biimpi 216	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ (◡𝐹 “ 𝐶))
27		fvimacnvi 7024	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (◡𝐹 “ 𝐶)) → (𝐹‘𝑥) ∈ 𝐶)
28	3, 26, 27	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ 𝐶)
29	24, 28	elind 4163	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ (ran 𝐹 ∩ 𝐶))
30	29, 10	eleqtrrdi 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ 𝐸)
31	30	fvresd 6878	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐺 ↾ 𝐸)‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
32	19, 31	eqtrid 2776	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑌‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
33	18, 32	eqtrd 2764	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑌‘(𝑋‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
34	2, 10, 7, 11	fcoreslem3 47066	. . . . . 6 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
35		fof 6772	. . . . . 6 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋:𝑃⟶𝐸)
36	34, 35	syl 17	. . . . 5 ⊢ (𝜑 → 𝑋:𝑃⟶𝐸)
37	36	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑋:𝑃⟶𝐸)
38	37, 15	fvco3d 6961	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝑌 ∘ 𝑋)‘𝑥) = (𝑌‘(𝑋‘𝑥)))
39	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝐹:𝐴⟶𝐵)
40	21	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑃 ⊆ dom 𝐹)
41	40	sselda 3946	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ dom 𝐹)
42	2	fdmd 6698	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
43	42	eqcomd 2735	. . . . . . 7 ⊢ (𝜑 → 𝐴 = dom 𝐹)
44	43	eleq2d 2814	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ dom 𝐹))
45	44	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ dom 𝐹))
46	41, 45	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ 𝐴)
47	39, 46	fvco3d 6961	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
48	33, 38, 47	3eqtr4rd 2775	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑥) = ((𝑌 ∘ 𝑋)‘𝑥))
49	9, 13, 48	eqfnfvd 7006	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3913   ⊆ wss 3914  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641   ∘ ccom 5642  Fun wfun 6505   Fn wfn 6506  ⟶wf 6507  –onto→wfo 6509  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519

This theorem is referenced by:  fcoresf1lem  47069  fcoresf1b  47071  fcoresfo  47072  fcoresfob  47073