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 7170	. . . 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 6666	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
4		funimaexg 6580	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V)
5	3, 4	sylan 581	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V)
6	5	resiexd 7165	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ( I ↾ (𝐹 “ 𝐴)) ∈ V)
7	2, 6, 5	3jca 1129	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∈ V ∧ ( I ↾ (𝐹 “ 𝐴)) ∈ V ∧ (𝐹 “ 𝐴) ∈ V))
8		eqid 2737	. . . 4 ⊢ (𝐹 “ 𝐴) = (𝐹 “ 𝐴)
9		eqid 2737	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))
10		eqid 2737	. . . 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 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 1137	. . . . 5 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
14		eqidd 2738	. . . . 5 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝐴 = 𝐴)
15		simp3 1139	. . . . 5 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝑝 = (𝐹 “ 𝐴))
16	13, 14, 15	foeq123d 6768	. . . 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 2738	. . . . . 6 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝐵 = 𝐵)
20	17, 18, 19	f1eq123d 6767	. . . . 5 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → (ℎ:𝑝–1-1→𝐵 ↔ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵))
21	20	3adant1 1131	. . . 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 5814	. . . . . . 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 1133	. . . . 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 2748	. . . 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 1439	. . 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 3546	. 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430   ↦ cmpt 5167   I cid 5519   ↾ cres 5627   “ cima 5628   ∘ ccom 5629  Fun wfun 6487  ⟶wf 6489  –1-1→wf1 6490  –onto→wfo 6491  ‘cfv 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501

This theorem is referenced by: (None)