Theorem fundcmpsurinjpreimafv 48489

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 48488	. 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 3455	. . . . . . 7 ⊢ 𝑗 ∈ V
4		vex 3455	. . . . . . 7 ⊢ 𝑓 ∈ V
5	3, 4	coex 7942	. . . . . 6 ⊢ (𝑗 ∘ 𝑓) ∈ V
6		simprl1 1237	. . . . . . 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 1156	. . . . . . . . 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 6823	. . . . . . . . . 10 ⊢ (𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) → 𝑓:𝑃–1-1→(𝐹 “ 𝐴))
9	8	3ad2ant2 1152	. . . . . . . . 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 6791	. . . . . . . . 9 ⊢ ((𝑗:(𝐹 “ 𝐴)–1-1→𝐵 ∧ 𝑓:𝑃–1-1→(𝐹 “ 𝐴)) → (𝑗 ∘ 𝑓):𝑃–1-1→𝐵)
11	7, 9, 10	syl2anc 596	. . . . . . . 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 741	. . . . . . 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 785	. . . . . . 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 1146	. . . . . 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 6773	. . . . . . . 8 ⊢ (ℎ = (𝑗 ∘ 𝑓) → (ℎ:𝑃–1-1→𝐵 ↔ (𝑗 ∘ 𝑓):𝑃–1-1→𝐵))
16		coeq1 5835	. . . . . . . . 9 ⊢ (ℎ = (𝑗 ∘ 𝑓) → (ℎ ∘ 𝑔) = ((𝑗 ∘ 𝑓) ∘ 𝑔))
17	16	eqeq2d 2772	. . . . . . . 8 ⊢ (ℎ = (𝑗 ∘ 𝑓) → (𝐹 = (ℎ ∘ 𝑔) ↔ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)))
18	15, 17	3anbi23d 1467	. . . . . . 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 3552	. . . . . 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 69	. . . . 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 418	. . . 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 1967	. . 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 1950	. 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 16	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  ∃wrex 3087  Vcvv 3451  {csn 4584  ^◡ccnv 5650   “ cima 5654   ∘ ccom 5655  ⟶wf 6534  –1-1→wf1 6535  –onto→wfo 6536  –1-1-onto→wf1o 6537  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546

This theorem is used by:  fundcmpsurinj  48490