Theorem fundcmpsurinjALT 47522

Description: Alternate proof of fundcmpsurinj 47519, based on fundcmpsurinjimaid 47521: 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 7155	. . . 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 6654	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
4		funimaexg 6568	. . . . 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 7150	. . 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 2731	. . . 4 ⊢ (𝐹 “ 𝐴) = (𝐹 “ 𝐴)
9		eqid 2731	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))
10		eqid 2731	. . . 4 ⊢ ( I ↾ (𝐹 “ 𝐴)) = ( I ↾ (𝐹 “ 𝐴))
11	8, 9, 10	fundcmpsurinjimaid 47521	. . 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 2732	. . . . 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 6756	. . . 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 2732	. . . . . 6 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝐵 = 𝐵)
20	17, 18, 19	f1eq123d 6755	. . . . 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 5803	. . . . . . 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 2742	. . . 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 3553	. 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 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436   ↦ cmpt 5170   I cid 5508   ↾ cres 5616   “ cima 5617   ∘ ccom 5618  Fun wfun 6475  ⟶wf 6477  –1-1→wf1 6478  –onto→wfo 6479  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489

This theorem is referenced by: (None)