Theorem fcores 47530

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 6659	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
4		fcof 6678	. . . . 5 ⊢ ((𝐺:𝐶⟶𝐷 ∧ Fun 𝐹) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)⟶𝐷)
5	1, 3, 4	syl2anc 590	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)⟶𝐷)
6	5	ffnd 6656	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) Fn (◡𝐹 “ 𝐶))
7		fcores.p	. . . 4 ⊢ 𝑃 = (◡𝐹 “ 𝐶)
8	7	fneq2i 6583	. . 3 ⊢ ((𝐺 ∘ 𝐹) Fn 𝑃 ↔ (𝐺 ∘ 𝐹) Fn (◡𝐹 “ 𝐶))
9	6, 8	sylibr 235	. 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 47529	. 2 ⊢ (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃)
14	11	fveq1i 6828	. . . . . 6 ⊢ (𝑋‘𝑥) = ((𝐹 ↾ 𝑃)‘𝑥)
15		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ 𝑃)
16	15	fvresd 6847	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐹 ↾ 𝑃)‘𝑥) = (𝐹‘𝑥))
17	14, 16	eqtrid 2786	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑋‘𝑥) = (𝐹‘𝑥))
18	17	fveq2d 6831	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑌‘(𝑋‘𝑥)) = (𝑌‘(𝐹‘𝑥)))
19	12	fveq1i 6828	. . . . 5 ⊢ (𝑌‘(𝐹‘𝑥)) = ((𝐺 ↾ 𝐸)‘(𝐹‘𝑥))
20		cnvimass 6034	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝐶) ⊆ dom 𝐹
21	7, 20	eqsstri 3961	. . . . . . . . . 10 ⊢ 𝑃 ⊆ dom 𝐹
22	21	sseli 3911	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ dom 𝐹)
23		fvelrn 7017	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
24	3, 22, 23	syl2an 602	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ ran 𝐹)
25	7	eleq2i 2831	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑃 ↔ 𝑥 ∈ (◡𝐹 “ 𝐶))
26	25	biimpi 217	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ (◡𝐹 “ 𝐶))
27		fvimacnvi 6993	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (◡𝐹 “ 𝐶)) → (𝐹‘𝑥) ∈ 𝐶)
28	3, 26, 27	syl2an 602	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ 𝐶)
29	24, 28	elind 4129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ (ran 𝐹 ∩ 𝐶))
30	29, 10	eleqtrrdi 2850	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝐹‘𝑥) ∈ 𝐸)
31	30	fvresd 6847	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐺 ↾ 𝐸)‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
32	19, 31	eqtrid 2786	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑌‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
33	18, 32	eqtrd 2774	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑌‘(𝑋‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
34	2, 10, 7, 11	fcoreslem3 47528	. . . . . 6 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
35		fof 6739	. . . . . 6 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋:𝑃⟶𝐸)
36	34, 35	syl 17	. . . . 5 ⊢ (𝜑 → 𝑋:𝑃⟶𝐸)
37	36	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑋:𝑃⟶𝐸)
38	37, 15	fvco3d 6928	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝑌 ∘ 𝑋)‘𝑥) = (𝑌‘(𝑋‘𝑥)))
39	2	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝐹:𝐴⟶𝐵)
40	21	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑃 ⊆ dom 𝐹)
41	40	sselda 3915	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ dom 𝐹)
42	2	fdmd 6665	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
43	42	eqcomd 2745	. . . . . . 7 ⊢ (𝜑 → 𝐴 = dom 𝐹)
44	43	eleq2d 2825	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ dom 𝐹))
45	44	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ dom 𝐹))
46	41, 45	mpbird 258	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ 𝐴)
47	39, 46	fvco3d 6928	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
48	33, 38, 47	3eqtr4rd 2785	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑥) = ((𝑌 ∘ 𝑋)‘𝑥))
49	9, 13, 48	eqfnfvd 6974	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∩ cin 3882   ⊆ wss 3883  ^◡ccnv 5617  dom cdm 5618  ran crn 5619   ↾ cres 5620   “ cima 5621   ∘ ccom 5622  Fun wfun 6479   Fn wfn 6480  ⟶wf 6481  –onto→wfo 6483  ‘cfv 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fo 6491  df-fv 6493

This theorem is referenced by:  fcoresf1lem  47531  fcoresf1b  47533  fcoresfo  47534  fcoresfob  47535