Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fundcmpsurinj Structured version   Visualization version   GIF version

Theorem fundcmpsurinj 47334
Description: Every function 𝐹:𝐴𝐵 can be decomposed into a surjective and an injective function. (Contributed by AV, 13-Mar-2024.)
Assertion
Ref Expression
fundcmpsurinj ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑝(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)))
Distinct variable groups:   𝐴,𝑔,,𝑝   𝐵,𝑔,,𝑝   𝑔,𝐹,,𝑝   𝑔,𝑉
Allowed substitution hints:   𝑉(,𝑝)

Proof of Theorem fundcmpsurinj
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abrexexg 7984 . . . 4 (𝐴𝑉 → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∈ V)
21adantl 481 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∈ V)
3 fveq2 6907 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
43sneqd 4643 . . . . . . . 8 (𝑥 = 𝑦 → {(𝐹𝑥)} = {(𝐹𝑦)})
54imaeq2d 6080 . . . . . . 7 (𝑥 = 𝑦 → (𝐹 “ {(𝐹𝑥)}) = (𝐹 “ {(𝐹𝑦)}))
65eqeq2d 2746 . . . . . 6 (𝑥 = 𝑦 → (𝑧 = (𝐹 “ {(𝐹𝑥)}) ↔ 𝑧 = (𝐹 “ {(𝐹𝑦)})))
76cbvrexvw 3236 . . . . 5 (∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)}) ↔ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)}))
87abbii 2807 . . . 4 {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}
98fundcmpsurinjpreimafv 47333 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔(𝑔:𝐴onto→{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∧ :{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}–1-1𝐵𝐹 = (𝑔)))
10 foeq3 6819 . . . . 5 (𝑝 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} → (𝑔:𝐴onto𝑝𝑔:𝐴onto→{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}))
11 f1eq2 6801 . . . . 5 (𝑝 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} → (:𝑝1-1𝐵:{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}–1-1𝐵))
1210, 113anbi12d 1436 . . . 4 (𝑝 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} → ((𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)) ↔ (𝑔:𝐴onto→{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∧ :{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}–1-1𝐵𝐹 = (𝑔))))
13122exbidv 1922 . . 3 (𝑝 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} → (∃𝑔(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)) ↔ ∃𝑔(𝑔:𝐴onto→{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∧ :{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}–1-1𝐵𝐹 = (𝑔))))
142, 9, 13spcedv 3598 . 2 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑝𝑔(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)))
15 exrot3 2163 . 2 (∃𝑝𝑔(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)) ↔ ∃𝑔𝑝(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)))
1614, 15sylib 218 1 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑝(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wrex 3068  Vcvv 3478  {csn 4631  ccnv 5688  cima 5692  ccom 5693  wf 6559  1-1wf1 6560  ontowfo 6561  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)
  Copyright terms: Public domain W3C validator