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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:𝐴𝐵𝐴𝑉) → ∃𝑔𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
Distinct variable groups:   𝐴,𝑔,,𝑝   𝐵,𝑔,,𝑝   𝑔,𝐹,,𝑝   𝑔,𝑉   𝐴,𝑖,𝑞,𝑔,,𝑝   𝐵,𝑖,𝑞   𝑖,𝐹,𝑞
Allowed substitution hints:   𝑉(,𝑖,𝑞,𝑝)

Proof of Theorem fundcmpsurbijinj
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffun 6501 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
2 funimaexg 6421 . . . 4 ((Fun 𝐹𝐴𝑉) → (𝐹𝐴) ∈ V)
31, 2sylan 583 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → (𝐹𝐴) ∈ V)
4 abrexexg 7666 . . . 4 (𝐴𝑉 → {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V)
54adantl 485 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V)
6 fveq2 6658 . . . . . . . . 9 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
76sneqd 4534 . . . . . . . 8 (𝑦 = 𝑥 → {(𝐹𝑦)} = {(𝐹𝑥)})
87imaeq2d 5901 . . . . . . 7 (𝑦 = 𝑥 → (𝐹 “ {(𝐹𝑦)}) = (𝐹 “ {(𝐹𝑥)}))
98eqeq2d 2769 . . . . . 6 (𝑦 = 𝑥 → (𝑧 = (𝐹 “ {(𝐹𝑦)}) ↔ 𝑧 = (𝐹 “ {(𝐹𝑥)})))
109cbvrexvw 3362 . . . . 5 (∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)}) ↔ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)}))
1110abbii 2823 . . . 4 {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1211fundcmpsurbijinjpreimafv 44314 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑖((𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ :{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
13 foeq3 6574 . . . . . . . . 9 (𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} → (𝑔:𝐴onto𝑝𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}))
1413adantl 485 . . . . . . . 8 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (𝑔:𝐴onto𝑝𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}))
15 f1oeq23 6593 . . . . . . . . 9 ((𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ 𝑞 = (𝐹𝐴)) → (:𝑝1-1-onto𝑞:{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴)))
1615ancoms 462 . . . . . . . 8 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (:𝑝1-1-onto𝑞:{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴)))
17 f1eq2 6556 . . . . . . . . 9 (𝑞 = (𝐹𝐴) → (𝑖:𝑞1-1𝐵𝑖:(𝐹𝐴)–1-1𝐵))
1817adantr 484 . . . . . . . 8 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (𝑖:𝑞1-1𝐵𝑖:(𝐹𝐴)–1-1𝐵))
1914, 16, 183anbi123d 1433 . . . . . . 7 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → ((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ↔ (𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ :{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵)))
2019anbi1d 632 . . . . . 6 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ((𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ :{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔))))
21203exbidv 1926 . . . . 5 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (∃𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ∃𝑔𝑖((𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ :{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔))))
2221spc2egv 3518 . . . 4 (((𝐹𝐴) ∈ V ∧ {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V) → (∃𝑔𝑖((𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ :{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) → ∃𝑞𝑝𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔))))
2322imp 410 . . 3 ((((𝐹𝐴) ∈ V ∧ {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V) ∧ ∃𝑔𝑖((𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ :{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔))) → ∃𝑞𝑝𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
243, 5, 12, 23syl21anc 836 . 2 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑞𝑝𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
25 exrot4 2170 . . 3 (∃𝑞𝑝𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ∃𝑔𝑞𝑝𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
26 excom13 2168 . . . 4 (∃𝑞𝑝𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ∃𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
27262exbii 1850 . . 3 (∃𝑔𝑞𝑝𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ∃𝑔𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
2825, 27bitri 278 . 2 (∃𝑞𝑝𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ∃𝑔𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
2924, 28sylib 221 1 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
 Colors of variables: wff setvar class 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)
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