Theorem fundcmpsurinjimaid 48311

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 6798	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴))
2		fundcmpsurinjimaid.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
3		ffn 6702	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
4		dffn5 6936	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
5	3, 4	sylib 221	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
6	5	eqcomd 2766	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = 𝐹)
7	2, 6	eqtrid 2807	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐺 = 𝐹)
8		eqidd 2761	. . . 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 6810	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐺:𝐴–onto→𝐼 ↔ 𝐹:𝐴–onto→(𝐹 “ 𝐴)))
12	1, 11	mpbird 260	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐺:𝐴–onto→𝐼)
13		f1oi 6856	. . 3 ⊢ ( I ↾ 𝐼):𝐼–1-1-onto→𝐼
14		f1of1 6816	. . 3 ⊢ (( I ↾ 𝐼):𝐼–1-1-onto→𝐼 → ( I ↾ 𝐼):𝐼–1-1→𝐼)
15		fundcmpsurinjimaid.h	. . . . . . 7 ⊢ 𝐻 = ( I ↾ 𝐼)
16		f1eq1 6766	. . . . . . 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 231	. . . . 5 ⊢ (( I ↾ 𝐼):𝐼–1-1→𝐼 → 𝐻:𝐼–1-1→𝐼)
19		fimass 6723	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵)
20	9, 19	eqsstrid 3969	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐼 ⊆ 𝐵)
21		f1ss 6778	. . . . 5 ⊢ ((𝐻:𝐼–1-1→𝐼 ∧ 𝐼 ⊆ 𝐵) → 𝐻:𝐼–1-1→𝐵)
22	18, 20, 21	syl2an 608	. . . 4 ⊢ ((( I ↾ 𝐼):𝐼–1-1→𝐼 ∧ 𝐹:𝐴⟶𝐵) → 𝐻:𝐼–1-1→𝐵)
23	22	ex 418	. . 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 6879	. . . . 5 ⊢ (𝐻‘(𝐹‘𝑥)) = (( I ↾ 𝐼)‘(𝐹‘𝑥))
26	3	adantr 486	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn 𝐴)
27		simpr 490	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
28	26, 27, 27	fnfvimad 7233	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))
29	28, 9	eleqtrrdi 2871	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐼)
30		fvresi 7171	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ 𝐼 → (( I ↾ 𝐼)‘(𝐹‘𝑥)) = (𝐹‘𝑥))
31	29, 30	syl 18	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (( I ↾ 𝐼)‘(𝐹‘𝑥)) = (𝐹‘𝑥))
32	25, 31	eqtrid 2807	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘(𝐹‘𝑥)) = (𝐹‘𝑥))
33	32	mpteq2dva 5198	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
34	2	coeq2i 5840	. . . 4 ⊢ (𝐻 ∘ 𝐺) = (𝐻 ∘ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
35		f1of 6817	. . . . . . . 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 6693	. . . . . . 7 ⊢ (𝐻:𝐼⟶𝐼 ↔ ( I ↾ 𝐼):𝐼⟶𝐼)
38	36, 37	mpbir 234	. . . . . 6 ⊢ 𝐻:𝐼⟶𝐼
39	38	a1i 11	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐻:𝐼⟶𝐼)
40	39, 29	cofmpt 7126	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝐻 ∘ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑥))))
41	34, 40	eqtrid 2807	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐻 ∘ 𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑥))))
42	33, 41, 5	3eqtr4rd 2806	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝐻 ∘ 𝐺))
43	12, 24, 42	3jca 1146	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐺:𝐴–onto→𝐼 ∧ 𝐻:𝐼–1-1→𝐵 ∧ 𝐹 = (𝐻 ∘ 𝐺)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899   ↦ cmpt 5186   I cid 5549   ↾ cres 5657   “ cima 5658   ∘ ccom 5659   Fn wfn 6528  ⟶wf 6529  –1-1→wf1 6530  –onto→wfo 6531  –1-1-onto→wf1o 6532  ‘cfv 6533

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 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  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 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541

This theorem is used by:  fundcmpsurinjALT  48312