Theorem fundcmpsurinjimaid 48042

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 6799	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴))
2		fundcmpsurinjimaid.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
3		ffn 6703	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
4		dffn5 6937	. . . . . . 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 2775	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = 𝐹)
7	2, 6	eqtrid 2816	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐺 = 𝐹)
8		eqidd 2770	. . . 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 6811	. . 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 6857	. . 3 ⊢ ( I ↾ 𝐼):𝐼–1-1-onto→𝐼
14		f1of1 6817	. . 3 ⊢ (( I ↾ 𝐼):𝐼–1-1-onto→𝐼 → ( I ↾ 𝐼):𝐼–1-1→𝐼)
15		fundcmpsurinjimaid.h	. . . . . . 7 ⊢ 𝐻 = ( I ↾ 𝐼)
16		f1eq1 6767	. . . . . . 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 6724	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵)
20	9, 19	eqsstrid 3983	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐼 ⊆ 𝐵)
21		f1ss 6779	. . . . 5 ⊢ ((𝐻:𝐼–1-1→𝐼 ∧ 𝐼 ⊆ 𝐵) → 𝐻:𝐼–1-1→𝐵)
22	18, 20, 21	syl2an 607	. . . 4 ⊢ ((( I ↾ 𝐼):𝐼–1-1→𝐼 ∧ 𝐹:𝐴⟶𝐵) → 𝐻:𝐼–1-1→𝐵)
23	22	ex 417	. . 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 6880	. . . . 5 ⊢ (𝐻‘(𝐹‘𝑥)) = (( I ↾ 𝐼)‘(𝐹‘𝑥))
26	3	adantr 485	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn 𝐴)
27		simpr 489	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
28	26, 27, 27	fnfvimad 7230	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))
29	28, 9	eleqtrrdi 2880	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐼)
30		fvresi 7169	. . . . . 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 2816	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘(𝐹‘𝑥)) = (𝐹‘𝑥))
33	32	mpteq2dva 5205	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
34	2	coeq2i 5844	. . . 4 ⊢ (𝐻 ∘ 𝐺) = (𝐻 ∘ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
35		f1of 6818	. . . . . . . 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 6694	. . . . . . 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 2816	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐻 ∘ 𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑥))))
42	33, 41, 5	3eqtr4rd 2815	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝐻 ∘ 𝐺))
43	12, 24, 42	3jca 1144	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐺:𝐴–onto→𝐼 ∧ 𝐻:𝐼–1-1→𝐵 ∧ 𝐹 = (𝐻 ∘ 𝐺)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913   ↦ cmpt 5193   I cid 5553   ↾ cres 5661   “ cima 5662   ∘ ccom 5663   Fn wfn 6528  ⟶wf 6529  –1-1→wf1 6530  –onto→wfo 6531  –1-1-onto→wf1o 6532  ‘cfv 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  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 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541

This theorem is referenced by:  fundcmpsurinjALT  48043