Theorem fundcmpsurinjALT 45916

Description: Alternate proof of fundcmpsurinj 45913, based on fundcmpsurinjimaid 45915: 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 7208	. . . 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 6708	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
4		funimaexg 6624	. . . . 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 7203	. . 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 2732	. . . 4 ⊢ (𝐹 “ 𝐴) = (𝐹 “ 𝐴)
9		eqid 2732	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))
10		eqid 2732	. . . 4 ⊢ ( I ↾ (𝐹 “ 𝐴)) = ( I ↾ (𝐹 “ 𝐴))
11	8, 9, 10	fundcmpsurinjimaid 45915	. . 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 1136	. . . . 5 ⊢ ((𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ∧ ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝑔 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
14		eqidd 2733	. . . . 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 6814	. . . 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 2733	. . . . . 6 ⊢ ((ℎ = ( I ↾ (𝐹 “ 𝐴)) ∧ 𝑝 = (𝐹 “ 𝐴)) → 𝐵 = 𝐵)
20	17, 18, 19	f1eq123d 6813	. . . . 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 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 5857	. . . . . . 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 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 2743	. . . 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 1436	. . 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 3591	. 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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3474   ↦ cmpt 5225   I cid 5567   ↾ cres 5672   “ cima 5673   ∘ ccom 5674  Fun wfun 6527  ⟶wf 6529  –1-1→wf1 6530  –onto→wfo 6531  ‘cfv 6533

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  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: (None)