Theorem fundcmpsurinj 47869

Description: Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective and an injective function. (Contributed by AV, 13-Mar-2024.)

Assertion

Ref	Expression
fundcmpsurinj	⊢ ((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   𝑔,𝑉

Allowed substitution hints:   𝑉(ℎ,𝑝)

Proof of Theorem fundcmpsurinj

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		abrexexg 7914	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∈ V)
2	1	adantl 481	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∈ V)
3		fveq2 6840	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
4	3	sneqd 4579	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {(𝐹‘𝑥)} = {(𝐹‘𝑦)})
5	4	imaeq2d 6025	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (◡𝐹 “ {(𝐹‘𝑥)}) = (◡𝐹 “ {(𝐹‘𝑦)}))
6	5	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})))
7	6	cbvrexvw 3216	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)}))
8	7	abbii 2803	. . . 4 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}
9	8	fundcmpsurinjpreimafv 47868	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ(𝑔:𝐴–onto→{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∧ ℎ:{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
10		foeq3 6750	. . . . 5 ⊢ (𝑝 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} → (𝑔:𝐴–onto→𝑝 ↔ 𝑔:𝐴–onto→{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}))
11		f1eq2 6732	. . . . 5 ⊢ (𝑝 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} → (ℎ:𝑝–1-1→𝐵 ↔ ℎ:{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}–1-1→𝐵))
12	10, 11	3anbi12d 1440	. . . 4 ⊢ (𝑝 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} → ((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)) ↔ (𝑔:𝐴–onto→{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∧ ℎ:{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))))
13	12	2exbidv 1926	. . 3 ⊢ (𝑝 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} → (∃𝑔∃ℎ(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)) ↔ ∃𝑔∃ℎ(𝑔:𝐴–onto→{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∧ ℎ:{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))))
14	2, 9, 13	spcedv 3540	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑝∃𝑔∃ℎ(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
15		exrot3 2171	. 2 ⊢ (∃𝑝∃𝑔∃ℎ(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)) ↔ ∃𝑔∃ℎ∃𝑝(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
16	14, 15	sylib 218	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   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2714  ∃wrex 3061  Vcvv 3429  {csn 4567  ^◡ccnv 5630   “ cima 5634   ∘ ccom 5635  ⟶wf 6494  –1-1→wf1 6495  –onto→wfo 6496  ‘cfv 6498

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506

This theorem is referenced by: (None)