Theorem fundcmpsurinjpreimafv 46066

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 46065	. 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 3478	. . . . . . 7 ⊢ 𝑗 ∈ V
4		vex 3478	. . . . . . 7 ⊢ 𝑓 ∈ V
5	3, 4	coex 7920	. . . . . 6 ⊢ (𝑗 ∘ 𝑓) ∈ V
6		simprl1 1218	. . . . . . 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 6832	. . . . . . . . . 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 6799	. . . . . . . . 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 726	. . . . . . 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 771	. . . . . . 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 6782	. . . . . . . 8 ⊢ (ℎ = (𝑗 ∘ 𝑓) → (ℎ:𝑃–1-1→𝐵 ↔ (𝑗 ∘ 𝑓):𝑃–1-1→𝐵))
16		coeq1 5857	. . . . . . . . 9 ⊢ (ℎ = (𝑗 ∘ 𝑓) → (ℎ ∘ 𝑔) = ((𝑗 ∘ 𝑓) ∘ 𝑔))
17	16	eqeq2d 2743	. . . . . . . 8 ⊢ (ℎ = (𝑗 ∘ 𝑓) → (𝐹 = (ℎ ∘ 𝑔) ↔ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)))
18	15, 17	3anbi23d 1439	. . . . . . 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 3587	. . . . . 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 413	. . . 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 1937	. . 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 1920	. 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 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2709  ∃wrex 3070  Vcvv 3474  {csn 4628  ^◡ccnv 5675   “ cima 5679   ∘ ccom 5680  ⟶wf 6539  –1-1→wf1 6540  –onto→wfo 6541  –1-1-onto→wf1o 6542  ‘cfv 6543

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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

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-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551

This theorem is referenced by:  fundcmpsurinj  46067