Theorem fundcmpsurinjALT 47386

Description: Alternate proof of fundcmpsurinj 47383, based on fundcmpsurinjimaid 47385: 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 7177	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∈ V)
2	1	adantl 481	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∈ V)
3		ffun 6673	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
4		funimaexg 6587	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V)
5	3, 4	sylan 580	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V)
6	5	resiexd 7172	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ( I ↾ (𝐹 “ 𝐴)) ∈ V)
7	2, 6, 5	3jca 1128	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∈ V ∧ ( I ↾ (𝐹 “ 𝐴)) ∈ V ∧ (𝐹 “ 𝐴) ∈ V))
8		eqid 2729	. . . 4 ⊢ (𝐹 “ 𝐴) = (𝐹 “ 𝐴)
9		eqid 2729	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))
10		eqid 2729	. . . 4 ⊢ ( I ↾ (𝐹 “ 𝐴)) = ( I ↾ (𝐹 “ 𝐴))
11	8, 9, 10	fundcmpsurinjimaid 47385	. . 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 480	. 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 1136	. . . . 5 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
14		eqidd 2730	. . . . 5 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝐴 = 𝐴)
15		simp3 1138	. . . . 5 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝑝 = (𝐹 “ 𝐴))
16	13, 14, 15	foeq123d 6775	. . . 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 482	. . . . . 6 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → ℎ = ( I ↾ (𝐹 “ 𝐴)))
18		simpr 484	. . . . . 6 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝑝 = (𝐹 “ 𝐴))
19		eqidd 2730	. . . . . 6 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝐵 = 𝐵)
20	17, 18, 19	f1eq123d 6774	. . . . 5 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → (ℎ:𝑝–1-1→𝐵 ↔ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵))
21	20	3adant1 1130	. . . 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 482	. . . . . . . 8 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))) → ℎ = ( I ↾ (𝐹 “ 𝐴)))
23		simpr 484	. . . . . . . 8 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))) → 𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
24	22, 23	coeq12d 5818	. . . . . . 7 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))) → (ℎ ∘ 𝑔) = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))))
25	24	ancoms 458	. . . . . 6 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴))) → (ℎ ∘ 𝑔) = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))))
26	25	3adant3 1132	. . . . 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 2740	. . . 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 1438	. . 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 3566	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3444   ↦ cmpt 5183   I cid 5525   ↾ cres 5633   “ cima 5634   ∘ ccom 5635  Fun wfun 6493  ⟶wf 6495  –1-1→wf1 6496  –onto→wfo 6497  ‘cfv 6499

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507

This theorem is referenced by: (None)