Theorem fundcmpsurinjALT 47887

Description: Alternate proof of fundcmpsurinj 47884, based on fundcmpsurinjimaid 47886: Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective and an injective function. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by AV, 13-Mar-2024.)

Assertion

Ref	Expression
fundcmpsurinjALT	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑝(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))

Distinct variable groups:   𝐴,𝑔,ℎ,𝑝   𝐵,𝑔,ℎ,𝑝   𝑔,𝐹,ℎ,𝑝

Allowed substitution hints:   𝑉(𝑔,ℎ,𝑝)

Proof of Theorem fundcmpsurinjALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mptexg 7165	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∈ V)
2	1	adantl 482	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∈ V)
3		ffun 6658	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
4		funimaexg 6572	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V)
5	3, 4	sylan 586	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V)
6	5	resiexd 7160	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ( I ↾ (𝐹 “ 𝐴)) ∈ V)
7	2, 6, 5	3jca 1134	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∈ V ∧ ( I ↾ (𝐹 “ 𝐴)) ∈ V ∧ (𝐹 “ 𝐴) ∈ V))
8		eqid 2739	. . . 4 ⊢ (𝐹 “ 𝐴) = (𝐹 “ 𝐴)
9		eqid 2739	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))
10		eqid 2739	. . . 4 ⊢ ( I ↾ (𝐹 “ 𝐴)) = ( I ↾ (𝐹 “ 𝐴))
11	8, 9, 10	fundcmpsurinjimaid 47886	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)):𝐴–onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵 ∧ 𝐹 = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))))
12	11	adantr 481	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)):𝐴–onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵 ∧ 𝐹 = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))))
13		simp1 1142	. . . . 5 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
14		eqidd 2740	. . . . 5 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝐴 = 𝐴)
15		simp3 1144	. . . . 5 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝑝 = (𝐹 “ 𝐴))
16	13, 14, 15	foeq123d 6760	. . . 4 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → (𝑔:𝐴–onto→𝑝 ↔ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)):𝐴–onto→(𝐹 “ 𝐴)))
17		simpl 483	. . . . . 6 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → ℎ = ( I ↾ (𝐹 “ 𝐴)))
18		simpr 485	. . . . . 6 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝑝 = (𝐹 “ 𝐴))
19		eqidd 2740	. . . . . 6 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝐵 = 𝐵)
20	17, 18, 19	f1eq123d 6759	. . . . 5 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → (ℎ:𝑝–1-1→𝐵 ↔ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵))
21	20	3adant1 1136	. . . 4 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → (ℎ:𝑝–1-1→𝐵 ↔ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵))
22		simpl 483	. . . . . . . 8 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))) → ℎ = ( I ↾ (𝐹 “ 𝐴)))
23		simpr 485	. . . . . . . 8 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))) → 𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
24	22, 23	coeq12d 5806	. . . . . . 7 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))) → (ℎ ∘ 𝑔) = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))))
25	24	ancoms 459	. . . . . 6 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴))) → (ℎ ∘ 𝑔) = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))))
26	25	3adant3 1138	. . . . 5 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → (ℎ ∘ 𝑔) = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))))
27	26	eqeq2d 2750	. . . 4 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → (𝐹 = (ℎ ∘ 𝑔) ↔ 𝐹 = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))))
28	16, 21, 27	3anbi123d 1444	. . 3 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → ((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)) ↔ ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)):𝐴–onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵 ∧ 𝐹 = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))))))
29	28	spc3egv 3541	. 2 ⊢ (((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∈ V ∧ ( I ↾ (𝐹 “ 𝐴)) ∈ V ∧ (𝐹 “ 𝐴) ∈ V) → (((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)):𝐴–onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵 ∧ 𝐹 = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))) → ∃𝑔∃ℎ∃𝑝(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))))
30	7, 12, 29	sylc 65	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑝(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3431   ↦ cmpt 5153   I cid 5512   ↾ cres 5620   “ cima 5621   ∘ ccom 5622  Fun wfun 6479  ⟶wf 6481  –1-1→wf1 6482  –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-rep 5199  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-reu 3345  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-iun 4923  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-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493

This theorem is referenced by: (None)