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 47396
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 7985 . . . 4 (𝐴𝑉 → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∈ V)
21adantl 481 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∈ V)
3 fveq2 6906 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
43sneqd 4638 . . . . . . . 8 (𝑥 = 𝑦 → {(𝐹𝑥)} = {(𝐹𝑦)})
54imaeq2d 6078 . . . . . . 7 (𝑥 = 𝑦 → (𝐹 “ {(𝐹𝑥)}) = (𝐹 “ {(𝐹𝑦)}))
65eqeq2d 2748 . . . . . 6 (𝑥 = 𝑦 → (𝑧 = (𝐹 “ {(𝐹𝑥)}) ↔ 𝑧 = (𝐹 “ {(𝐹𝑦)})))
76cbvrexvw 3238 . . . . 5 (∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)}) ↔ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)}))
87abbii 2809 . . . 4 {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}
98fundcmpsurinjpreimafv 47395 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔(𝑔:𝐴onto→{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∧ :{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}–1-1𝐵𝐹 = (𝑔)))
10 foeq3 6818 . . . . 5 (𝑝 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} → (𝑔:𝐴onto𝑝𝑔:𝐴onto→{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}))
11 f1eq2 6800 . . . . 5 (𝑝 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} → (:𝑝1-1𝐵:{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}–1-1𝐵))
1210, 113anbi12d 1439 . . . 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 1924 . . 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 2165 . 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 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480  {csn 4626  ccnv 5684  cima 5688  ccom 5689  wf 6557  1-1wf1 6558  ontowfo 6559  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)
  Copyright terms: Public domain W3C validator