Theorem fundcmpsurbijinj 44317

Description: Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective, a bijective and an injective function. (Contributed by AV, 23-Mar-2024.)

Assertion

Ref	Expression
fundcmpsurbijinj	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖∃𝑝∃𝑞((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))

Distinct variable groups:   𝐴,𝑔,ℎ,𝑝   𝐵,𝑔,ℎ,𝑝   𝑔,𝐹,ℎ,𝑝   𝑔,𝑉   𝐴,𝑖,𝑞,𝑔,ℎ,𝑝   𝐵,𝑖,𝑞   𝑖,𝐹,𝑞

Allowed substitution hints:   𝑉(ℎ,𝑖,𝑞,𝑝)

Proof of Theorem fundcmpsurbijinj

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

Step	Hyp	Ref	Expression
1		ffun 6501	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
2		funimaexg 6421	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V)
3	1, 2	sylan 583	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V)
4		abrexexg 7666	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∈ V)
5	4	adantl 485	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∈ V)
6		fveq2 6658	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
7	6	sneqd 4534	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → {(𝐹‘𝑦)} = {(𝐹‘𝑥)})
8	7	imaeq2d 5901	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (◡𝐹 “ {(𝐹‘𝑦)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
9	8	eqeq2d 2769	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑧 = (◡𝐹 “ {(𝐹‘𝑦)}) ↔ 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})))
10	9	cbvrexvw 3362	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)}) ↔ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}))
11	10	abbii 2823	. . . 4 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
12	11	fundcmpsurbijinjpreimafv 44314	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
13		foeq3 6574	. . . . . . . . 9 ⊢ (𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} → (𝑔:𝐴–onto→𝑝 ↔ 𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}))
14	13	adantl 485	. . . . . . . 8 ⊢ ((𝑞 = (𝐹 “ 𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}) → (𝑔:𝐴–onto→𝑝 ↔ 𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}))
15		f1oeq23 6593	. . . . . . . . 9 ⊢ ((𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ 𝑞 = (𝐹 “ 𝐴)) → (ℎ:𝑝–1-1-onto→𝑞 ↔ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴)))
16	15	ancoms 462	. . . . . . . 8 ⊢ ((𝑞 = (𝐹 “ 𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}) → (ℎ:𝑝–1-1-onto→𝑞 ↔ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴)))
17		f1eq2 6556	. . . . . . . . 9 ⊢ (𝑞 = (𝐹 “ 𝐴) → (𝑖:𝑞–1-1→𝐵 ↔ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵))
18	17	adantr 484	. . . . . . . 8 ⊢ ((𝑞 = (𝐹 “ 𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}) → (𝑖:𝑞–1-1→𝐵 ↔ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵))
19	14, 16, 18	3anbi123d 1433	. . . . . . 7 ⊢ ((𝑞 = (𝐹 “ 𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}) → ((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ↔ (𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵)))
20	19	anbi1d 632	. . . . . 6 ⊢ ((𝑞 = (𝐹 “ 𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}) → (((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ ((𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))))
21	20	3exbidv 1926	. . . . 5 ⊢ ((𝑞 = (𝐹 “ 𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}) → (∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ ∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))))
22	21	spc2egv 3518	. . . 4 ⊢ (((𝐹 “ 𝐴) ∈ V ∧ {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∈ V) → (∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) → ∃𝑞∃𝑝∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))))
23	22	imp 410	. . 3 ⊢ ((((𝐹 “ 𝐴) ∈ V ∧ {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∈ V) ∧ ∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))) → ∃𝑞∃𝑝∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
24	3, 5, 12, 23	syl21anc 836	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑞∃𝑝∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
25		exrot4 2170	. . 3 ⊢ (∃𝑞∃𝑝∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ ∃𝑔∃ℎ∃𝑞∃𝑝∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
26		excom13 2168	. . . 4 ⊢ (∃𝑞∃𝑝∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ ∃𝑖∃𝑝∃𝑞((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
27	26	2exbii 1850	. . 3 ⊢ (∃𝑔∃ℎ∃𝑞∃𝑝∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ ∃𝑔∃ℎ∃𝑖∃𝑝∃𝑞((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
28	25, 27	bitri 278	. 2 ⊢ (∃𝑞∃𝑝∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ ∃𝑔∃ℎ∃𝑖∃𝑝∃𝑞((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
29	24, 28	sylib 221	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖∃𝑝∃𝑞((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735  ∃wrex 3071  Vcvv 3409  {csn 4522  ^◡ccnv 5523   “ cima 5527   ∘ ccom 5528  Fun wfun 6329  ⟶wf 6331  –1-1→wf1 6332  –onto→wfo 6333  –1-1-onto→wf1o 6334  ‘cfv 6335

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343

This theorem is referenced by: (None)