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Theorem fundcmpsurinjpreimafv 47282
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 47281 . 2 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑓𝑗((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔)))
3 vex 3492 . . . . . . 7 𝑗 ∈ V
4 vex 3492 . . . . . . 7 𝑓 ∈ V
53, 4coex 7970 . . . . . 6 (𝑗𝑓) ∈ V
6 simprl1 1218 . . . . . . 7 (((𝐹:𝐴𝐵𝐴𝑉) ∧ ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔))) → 𝑔:𝐴onto𝑃)
7 simp3 1138 . . . . . . . . 9 ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) → 𝑗:(𝐹𝐴)–1-1𝐵)
8 f1of1 6861 . . . . . . . . . 10 (𝑓:𝑃1-1-onto→(𝐹𝐴) → 𝑓:𝑃1-1→(𝐹𝐴))
983ad2ant2 1134 . . . . . . . . 9 ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) → 𝑓:𝑃1-1→(𝐹𝐴))
10 f1co 6828 . . . . . . . . 9 ((𝑗:(𝐹𝐴)–1-1𝐵𝑓:𝑃1-1→(𝐹𝐴)) → (𝑗𝑓):𝑃1-1𝐵)
117, 9, 10syl2anc 583 . . . . . . . 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 1128 . . . . . 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 6812 . . . . . . . 8 ( = (𝑗𝑓) → (:𝑃1-1𝐵 ↔ (𝑗𝑓):𝑃1-1𝐵))
16 coeq1 5882 . . . . . . . . 9 ( = (𝑗𝑓) → (𝑔) = ((𝑗𝑓) ∘ 𝑔))
1716eqeq2d 2751 . . . . . . . 8 ( = (𝑗𝑓) → (𝐹 = (𝑔) ↔ 𝐹 = ((𝑗𝑓) ∘ 𝑔)))
1815, 173anbi23d 1439 . . . . . . 7 ( = (𝑗𝑓) → ((𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔)) ↔ (𝑔:𝐴onto𝑃 ∧ (𝑗𝑓):𝑃1-1𝐵𝐹 = ((𝑗𝑓) ∘ 𝑔))))
1918spcegv 3610 . . . . . 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 1933 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → (∃𝑓𝑗((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔)) → ∃(𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔))))
2322eximdv 1916 . 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 1087   = wceq 1537  wex 1777  wcel 2108  {cab 2717  wrex 3076  Vcvv 3488  {csn 4648  ccnv 5699  cima 5703  ccom 5704  wf 6569  1-1wf1 6570  ontowfo 6571  1-1-ontowf1o 6572  cfv 6573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581
This theorem is referenced by:  fundcmpsurinj  47283
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