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Theorem fundcmpsurbijinj 47397
Description: Every function 𝐹:𝐴𝐵 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 6739 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
2 funimaexg 6653 . . . 4 ((Fun 𝐹𝐴𝑉) → (𝐹𝐴) ∈ V)
31, 2sylan 580 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → (𝐹𝐴) ∈ V)
4 abrexexg 7985 . . . 4 (𝐴𝑉 → {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V)
54adantl 481 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V)
6 fveq2 6906 . . . . . . . . 9 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
76sneqd 4638 . . . . . . . 8 (𝑦 = 𝑥 → {(𝐹𝑦)} = {(𝐹𝑥)})
87imaeq2d 6078 . . . . . . 7 (𝑦 = 𝑥 → (𝐹 “ {(𝐹𝑦)}) = (𝐹 “ {(𝐹𝑥)}))
98eqeq2d 2748 . . . . . 6 (𝑦 = 𝑥 → (𝑧 = (𝐹 “ {(𝐹𝑦)}) ↔ 𝑧 = (𝐹 “ {(𝐹𝑥)})))
109cbvrexvw 3238 . . . . 5 (∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)}) ↔ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)}))
1110abbii 2809 . . . 4 {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1211fundcmpsurbijinjpreimafv 47394 . . 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 6818 . . . . . . . . 9 (𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} → (𝑔:𝐴onto𝑝𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}))
1413adantl 481 . . . . . . . 8 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (𝑔:𝐴onto𝑝𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}))
15 f1oeq23 6839 . . . . . . . . 9 ((𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ 𝑞 = (𝐹𝐴)) → (:𝑝1-1-onto𝑞:{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴)))
1615ancoms 458 . . . . . . . 8 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (:𝑝1-1-onto𝑞:{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴)))
17 f1eq2 6800 . . . . . . . . 9 (𝑞 = (𝐹𝐴) → (𝑖:𝑞1-1𝐵𝑖:(𝐹𝐴)–1-1𝐵))
1817adantr 480 . . . . . . . 8 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (𝑖:𝑞1-1𝐵𝑖:(𝐹𝐴)–1-1𝐵))
1914, 16, 183anbi123d 1438 . . . . . . 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 631 . . . . . 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 1925 . . . . 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 3599 . . . 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 406 . . 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 838 . 2 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑞𝑝𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
25 exrot4 2166 . . 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 2164 . . . 4 (∃𝑞𝑝𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ∃𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
27262exbii 1849 . . 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 275 . 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 218 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 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480  {csn 4626  ccnv 5684  cima 5688  ccom 5689  Fun wfun 6555  wf 6557  1-1wf1 6558  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by: (None)
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