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

Theorem fundcmpsurinjpreimafv 47333
Description: Every function 𝐹:𝐴𝐵 can be decomposed into a surjective function onto 𝑃 and an injective function from 𝑃. (Contributed by AV, 12-Mar-2024.) (Proof shortened by AV, 22-Mar-2024.)
Hypothesis
Ref Expression
fundcmpsurinj.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
fundcmpsurinjpreimafv ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔(𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔)))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝐴,𝑔,   𝐵,𝑔,,𝑥   𝑧,𝐵   𝑔,𝐹,   𝑃,𝑔,,𝑥   𝑥,𝑉,𝑔
Allowed substitution hints:   𝑃(𝑧)   𝑉(𝑧,)

Proof of Theorem fundcmpsurinjpreimafv
Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fundcmpsurinj.p . . 3 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
21fundcmpsurbijinjpreimafv 47332 . 2 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑓𝑗((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔)))
3 vex 3482 . . . . . . 7 𝑗 ∈ V
4 vex 3482 . . . . . . 7 𝑓 ∈ V
53, 4coex 7953 . . . . . 6 (𝑗𝑓) ∈ V
6 simprl1 1217 . . . . . . 7 (((𝐹:𝐴𝐵𝐴𝑉) ∧ ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔))) → 𝑔:𝐴onto𝑃)
7 simp3 1137 . . . . . . . . 9 ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) → 𝑗:(𝐹𝐴)–1-1𝐵)
8 f1of1 6848 . . . . . . . . . 10 (𝑓:𝑃1-1-onto→(𝐹𝐴) → 𝑓:𝑃1-1→(𝐹𝐴))
983ad2ant2 1133 . . . . . . . . 9 ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) → 𝑓:𝑃1-1→(𝐹𝐴))
10 f1co 6816 . . . . . . . . 9 ((𝑗:(𝐹𝐴)–1-1𝐵𝑓:𝑃1-1→(𝐹𝐴)) → (𝑗𝑓):𝑃1-1𝐵)
117, 9, 10syl2anc 584 . . . . . . . 8 ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) → (𝑗𝑓):𝑃1-1𝐵)
1211ad2antrl 728 . . . . . . 7 (((𝐹:𝐴𝐵𝐴𝑉) ∧ ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔))) → (𝑗𝑓):𝑃1-1𝐵)
13 simprr 773 . . . . . . 7 (((𝐹:𝐴𝐵𝐴𝑉) ∧ ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔))) → 𝐹 = ((𝑗𝑓) ∘ 𝑔))
146, 12, 133jca 1127 . . . . . 6 (((𝐹:𝐴𝐵𝐴𝑉) ∧ ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔))) → (𝑔:𝐴onto𝑃 ∧ (𝑗𝑓):𝑃1-1𝐵𝐹 = ((𝑗𝑓) ∘ 𝑔)))
15 f1eq1 6800 . . . . . . . 8 ( = (𝑗𝑓) → (:𝑃1-1𝐵 ↔ (𝑗𝑓):𝑃1-1𝐵))
16 coeq1 5871 . . . . . . . . 9 ( = (𝑗𝑓) → (𝑔) = ((𝑗𝑓) ∘ 𝑔))
1716eqeq2d 2746 . . . . . . . 8 ( = (𝑗𝑓) → (𝐹 = (𝑔) ↔ 𝐹 = ((𝑗𝑓) ∘ 𝑔)))
1815, 173anbi23d 1438 . . . . . . 7 ( = (𝑗𝑓) → ((𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔)) ↔ (𝑔:𝐴onto𝑃 ∧ (𝑗𝑓):𝑃1-1𝐵𝐹 = ((𝑗𝑓) ∘ 𝑔))))
1918spcegv 3597 . . . . . 6 ((𝑗𝑓) ∈ V → ((𝑔:𝐴onto𝑃 ∧ (𝑗𝑓):𝑃1-1𝐵𝐹 = ((𝑗𝑓) ∘ 𝑔)) → ∃(𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔))))
205, 14, 19mpsyl 68 . . . . 5 (((𝐹:𝐴𝐵𝐴𝑉) ∧ ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔))) → ∃(𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔)))
2120ex 412 . . . 4 ((𝐹:𝐴𝐵𝐴𝑉) → (((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔)) → ∃(𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔))))
2221exlimdvv 1932 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → (∃𝑓𝑗((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔)) → ∃(𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔))))
2322eximdv 1915 . 2 ((𝐹:𝐴𝐵𝐴𝑉) → (∃𝑔𝑓𝑗((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔)) → ∃𝑔(𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔))))
242, 23mpd 15 1 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔(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  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:  fundcmpsurinj  47334
  Copyright terms: Public domain W3C validator