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Theorem fundcmpsurbijinj 47335
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 6740 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
2 funimaexg 6654 . . . 4 ((Fun 𝐹𝐴𝑉) → (𝐹𝐴) ∈ V)
31, 2sylan 580 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → (𝐹𝐴) ∈ V)
4 abrexexg 7984 . . . 4 (𝐴𝑉 → {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V)
54adantl 481 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V)
6 fveq2 6907 . . . . . . . . 9 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
76sneqd 4643 . . . . . . . 8 (𝑦 = 𝑥 → {(𝐹𝑦)} = {(𝐹𝑥)})
87imaeq2d 6080 . . . . . . 7 (𝑦 = 𝑥 → (𝐹 “ {(𝐹𝑦)}) = (𝐹 “ {(𝐹𝑥)}))
98eqeq2d 2746 . . . . . 6 (𝑦 = 𝑥 → (𝑧 = (𝐹 “ {(𝐹𝑦)}) ↔ 𝑧 = (𝐹 “ {(𝐹𝑥)})))
109cbvrexvw 3236 . . . . 5 (∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)}) ↔ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)}))
1110abbii 2807 . . . 4 {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1211fundcmpsurbijinjpreimafv 47332 . . 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 6819 . . . . . . . . 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 6840 . . . . . . . . 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 6801 . . . . . . . . 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 1435 . . . . . . 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 1923 . . . . 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 2164 . . 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 2162 . . . 4 (∃𝑞𝑝𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ∃𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
27262exbii 1846 . . 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 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wrex 3068  Vcvv 3478  {csn 4631  ccnv 5688  cima 5692  ccom 5693  Fun wfun 6557  wf 6559  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by: (None)
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