Theorem fundcmpsurinjpreimafv 47664

Description: Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective function onto 𝑃 and an injective function from 𝑃. (Contributed by AV, 12-Mar-2024.) (Proof shortened by AV, 22-Mar-2024.)

Hypothesis

Ref	Expression
fundcmpsurinj.p	⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}

Assertion

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

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝐴,𝑔,ℎ   𝐵,𝑔,ℎ,𝑥   𝑧,𝐵   𝑔,𝐹,ℎ   𝑃,𝑔,ℎ,𝑥   𝑥,𝑉,𝑔

Allowed substitution hints:   𝑃(𝑧)   𝑉(𝑧,ℎ)

Proof of Theorem fundcmpsurinjpreimafv

Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fundcmpsurinj.p	. . 3 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2	1	fundcmpsurbijinjpreimafv 47663	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃𝑓∃𝑗((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)))
3		vex 3444	. . . . . . 7 ⊢ 𝑗 ∈ V
4		vex 3444	. . . . . . 7 ⊢ 𝑓 ∈ V
5	3, 4	coex 7872	. . . . . 6 ⊢ (𝑗 ∘ 𝑓) ∈ V
6		simprl1 1219	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))) → 𝑔:𝐴–onto→𝑃)
7		simp3 1138	. . . . . . . . 9 ⊢ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) → 𝑗:(𝐹 “ 𝐴)–1-1→𝐵)
8		f1of1 6773	. . . . . . . . . 10 ⊢ (𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) → 𝑓:𝑃–1-1→(𝐹 “ 𝐴))
9	8	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) → 𝑓:𝑃–1-1→(𝐹 “ 𝐴))
10		f1co 6741	. . . . . . . . 9 ⊢ ((𝑗:(𝐹 “ 𝐴)–1-1→𝐵 ∧ 𝑓:𝑃–1-1→(𝐹 “ 𝐴)) → (𝑗 ∘ 𝑓):𝑃–1-1→𝐵)
11	7, 9, 10	syl2anc 584	. . . . . . . 8 ⊢ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) → (𝑗 ∘ 𝑓):𝑃–1-1→𝐵)
12	11	ad2antrl 728	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))) → (𝑗 ∘ 𝑓):𝑃–1-1→𝐵)
13		simprr 772	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))) → 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))
14	6, 12, 13	3jca 1128	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))) → (𝑔:𝐴–onto→𝑃 ∧ (𝑗 ∘ 𝑓):𝑃–1-1→𝐵 ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)))
15		f1eq1 6725	. . . . . . . 8 ⊢ (ℎ = (𝑗 ∘ 𝑓) → (ℎ:𝑃–1-1→𝐵 ↔ (𝑗 ∘ 𝑓):𝑃–1-1→𝐵))
16		coeq1 5806	. . . . . . . . 9 ⊢ (ℎ = (𝑗 ∘ 𝑓) → (ℎ ∘ 𝑔) = ((𝑗 ∘ 𝑓) ∘ 𝑔))
17	16	eqeq2d 2747	. . . . . . . 8 ⊢ (ℎ = (𝑗 ∘ 𝑓) → (𝐹 = (ℎ ∘ 𝑔) ↔ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)))
18	15, 17	3anbi23d 1441	. . . . . . 7 ⊢ (ℎ = (𝑗 ∘ 𝑓) → ((𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)) ↔ (𝑔:𝐴–onto→𝑃 ∧ (𝑗 ∘ 𝑓):𝑃–1-1→𝐵 ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))))
19	18	spcegv 3551	. . . . . 6 ⊢ ((𝑗 ∘ 𝑓) ∈ V → ((𝑔:𝐴–onto→𝑃 ∧ (𝑗 ∘ 𝑓):𝑃–1-1→𝐵 ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)) → ∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))))
20	5, 14, 19	mpsyl 68	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))) → ∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
21	20	ex 412	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)) → ∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))))
22	21	exlimdvv 1935	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (∃𝑓∃𝑗((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)) → ∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))))
23	22	eximdv 1918	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (∃𝑔∃𝑓∃𝑗((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)) → ∃𝑔∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))))
24	2, 23	mpd 15	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714  ∃wrex 3060  Vcvv 3440  {csn 4580  ^◡ccnv 5623   “ cima 5627   ∘ ccom 5628  ⟶wf 6488  –1-1→wf1 6489  –onto→wfo 6490  –1-1-onto→wf1o 6491  ‘cfv 6492

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500

This theorem is referenced by:  fundcmpsurinj  47665