Theorem fundcmpsurinjimaid 47405

Description: Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective function onto the image (𝐹 “ 𝐴) of the domain of 𝐹 and an injective function from the image (𝐹 “ 𝐴). (Contributed by AV, 17-Mar-2024.)

Hypotheses

Ref	Expression
fundcmpsurinjimaid.i	⊢ 𝐼 = (𝐹 “ 𝐴)
fundcmpsurinjimaid.g	⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
fundcmpsurinjimaid.h	⊢ 𝐻 = ( I ↾ 𝐼)

Assertion

Ref	Expression
fundcmpsurinjimaid	⊢ (𝐹:𝐴⟶𝐵 → (𝐺:𝐴–onto→𝐼 ∧ 𝐻:𝐼–1-1→𝐵 ∧ 𝐹 = (𝐻 ∘ 𝐺)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐻   𝑥,𝐼

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem fundcmpsurinjimaid

Step	Hyp	Ref	Expression
1		fimadmfo 6804	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴))
2		fundcmpsurinjimaid.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
3		ffn 6711	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
4		dffn5 6942	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
5	3, 4	sylib 218	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
6	5	eqcomd 2742	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = 𝐹)
7	2, 6	eqtrid 2783	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐺 = 𝐹)
8		eqidd 2737	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = 𝐴)
9		fundcmpsurinjimaid.i	. . . . 5 ⊢ 𝐼 = (𝐹 “ 𝐴)
10	9	a1i 11	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐼 = (𝐹 “ 𝐴))
11	7, 8, 10	foeq123d 6816	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐺:𝐴–onto→𝐼 ↔ 𝐹:𝐴–onto→(𝐹 “ 𝐴)))
12	1, 11	mpbird 257	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐺:𝐴–onto→𝐼)
13		f1oi 6861	. . 3 ⊢ ( I ↾ 𝐼):𝐼–1-1-onto→𝐼
14		f1of1 6822	. . 3 ⊢ (( I ↾ 𝐼):𝐼–1-1-onto→𝐼 → ( I ↾ 𝐼):𝐼–1-1→𝐼)
15		fundcmpsurinjimaid.h	. . . . . . 7 ⊢ 𝐻 = ( I ↾ 𝐼)
16		f1eq1 6774	. . . . . . 7 ⊢ (𝐻 = ( I ↾ 𝐼) → (𝐻:𝐼–1-1→𝐼 ↔ ( I ↾ 𝐼):𝐼–1-1→𝐼))
17	15, 16	ax-mp 5	. . . . . 6 ⊢ (𝐻:𝐼–1-1→𝐼 ↔ ( I ↾ 𝐼):𝐼–1-1→𝐼)
18	17	biimpri 228	. . . . 5 ⊢ (( I ↾ 𝐼):𝐼–1-1→𝐼 → 𝐻:𝐼–1-1→𝐼)
19		fimass 6731	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵)
20	9, 19	eqsstrid 4002	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐼 ⊆ 𝐵)
21		f1ss 6784	. . . . 5 ⊢ ((𝐻:𝐼–1-1→𝐼 ∧ 𝐼 ⊆ 𝐵) → 𝐻:𝐼–1-1→𝐵)
22	18, 20, 21	syl2an 596	. . . 4 ⊢ ((( I ↾ 𝐼):𝐼–1-1→𝐼 ∧ 𝐹:𝐴⟶𝐵) → 𝐻:𝐼–1-1→𝐵)
23	22	ex 412	. . 3 ⊢ (( I ↾ 𝐼):𝐼–1-1→𝐼 → (𝐹:𝐴⟶𝐵 → 𝐻:𝐼–1-1→𝐵))
24	13, 14, 23	mp2b 10	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐻:𝐼–1-1→𝐵)
25	15	fveq1i 6882	. . . . 5 ⊢ (𝐻‘(𝐹‘𝑥)) = (( I ↾ 𝐼)‘(𝐹‘𝑥))
26	3	adantr 480	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn 𝐴)
27		simpr 484	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
28	26, 27, 27	fnfvimad 7231	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))
29	28, 9	eleqtrrdi 2846	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐼)
30		fvresi 7170	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ 𝐼 → (( I ↾ 𝐼)‘(𝐹‘𝑥)) = (𝐹‘𝑥))
31	29, 30	syl 17	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (( I ↾ 𝐼)‘(𝐹‘𝑥)) = (𝐹‘𝑥))
32	25, 31	eqtrid 2783	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘(𝐹‘𝑥)) = (𝐹‘𝑥))
33	32	mpteq2dva 5219	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
34	2	coeq2i 5845	. . . 4 ⊢ (𝐻 ∘ 𝐺) = (𝐻 ∘ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
35		f1of 6823	. . . . . . . 8 ⊢ (( I ↾ 𝐼):𝐼–1-1-onto→𝐼 → ( I ↾ 𝐼):𝐼⟶𝐼)
36	13, 35	ax-mp 5	. . . . . . 7 ⊢ ( I ↾ 𝐼):𝐼⟶𝐼
37	15	feq1i 6702	. . . . . . 7 ⊢ (𝐻:𝐼⟶𝐼 ↔ ( I ↾ 𝐼):𝐼⟶𝐼)
38	36, 37	mpbir 231	. . . . . 6 ⊢ 𝐻:𝐼⟶𝐼
39	38	a1i 11	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐻:𝐼⟶𝐼)
40	39, 29	cofmpt 7127	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝐻 ∘ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑥))))
41	34, 40	eqtrid 2783	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐻 ∘ 𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑥))))
42	33, 41, 5	3eqtr4rd 2782	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝐻 ∘ 𝐺))
43	12, 24, 42	3jca 1128	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐺:𝐴–onto→𝐼 ∧ 𝐻:𝐼–1-1→𝐵 ∧ 𝐹 = (𝐻 ∘ 𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3931   ↦ cmpt 5206   I cid 5552   ↾ cres 5661   “ cima 5662   ∘ ccom 5663   Fn wfn 6531  ⟶wf 6532  –1-1→wf1 6533  –onto→wfo 6534  –1-1-onto→wf1o 6535  ‘cfv 6536

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by:  fundcmpsurinjALT  47406