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 46076
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 46075 . 2 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑓𝑗((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔)))
3 vex 3479 . . . . . . 7 𝑗 ∈ V
4 vex 3479 . . . . . . 7 𝑓 ∈ V
53, 4coex 7921 . . . . . 6 (𝑗𝑓) ∈ V
6 simprl1 1219 . . . . . . 7 (((𝐹:𝐴𝐵𝐴𝑉) ∧ ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔))) → 𝑔:𝐴onto𝑃)
7 simp3 1139 . . . . . . . . 9 ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) → 𝑗:(𝐹𝐴)–1-1𝐵)
8 f1of1 6833 . . . . . . . . . 10 (𝑓:𝑃1-1-onto→(𝐹𝐴) → 𝑓:𝑃1-1→(𝐹𝐴))
983ad2ant2 1135 . . . . . . . . 9 ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) → 𝑓:𝑃1-1→(𝐹𝐴))
10 f1co 6800 . . . . . . . . 9 ((𝑗:(𝐹𝐴)–1-1𝐵𝑓:𝑃1-1→(𝐹𝐴)) → (𝑗𝑓):𝑃1-1𝐵)
117, 9, 10syl2anc 585 . . . . . . . 8 ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) → (𝑗𝑓):𝑃1-1𝐵)
1211ad2antrl 727 . . . . . . 7 (((𝐹:𝐴𝐵𝐴𝑉) ∧ ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔))) → (𝑗𝑓):𝑃1-1𝐵)
13 simprr 772 . . . . . . 7 (((𝐹:𝐴𝐵𝐴𝑉) ∧ ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔))) → 𝐹 = ((𝑗𝑓) ∘ 𝑔))
146, 12, 133jca 1129 . . . . . 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 6783 . . . . . . . 8 ( = (𝑗𝑓) → (:𝑃1-1𝐵 ↔ (𝑗𝑓):𝑃1-1𝐵))
16 coeq1 5858 . . . . . . . . 9 ( = (𝑗𝑓) → (𝑔) = ((𝑗𝑓) ∘ 𝑔))
1716eqeq2d 2744 . . . . . . . 8 ( = (𝑗𝑓) → (𝐹 = (𝑔) ↔ 𝐹 = ((𝑗𝑓) ∘ 𝑔)))
1815, 173anbi23d 1440 . . . . . . 7 ( = (𝑗𝑓) → ((𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔)) ↔ (𝑔:𝐴onto𝑃 ∧ (𝑗𝑓):𝑃1-1𝐵𝐹 = ((𝑗𝑓) ∘ 𝑔))))
1918spcegv 3588 . . . . . 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 414 . . . 4 ((𝐹:𝐴𝐵𝐴𝑉) → (((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔)) → ∃(𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔))))
2221exlimdvv 1938 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → (∃𝑓𝑗((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔)) → ∃(𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔))))
2322eximdv 1921 . 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 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wrex 3071  Vcvv 3475  {csn 4629  ccnv 5676  cima 5680  ccom 5681  wf 6540  1-1wf1 6541  ontowfo 6542  1-1-ontowf1o 6543  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552
This theorem is referenced by:  fundcmpsurinj  46077
  Copyright terms: Public domain W3C validator