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Theorem fundcmpsurinjpreimafv 44283
 Description: Every function 𝐹:𝐴⟶𝐵 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 44282 . 2 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑓𝑗((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔)))
3 vex 3414 . . . . . . 7 𝑗 ∈ V
4 vex 3414 . . . . . . 7 𝑓 ∈ V
53, 4coex 7638 . . . . . 6 (𝑗𝑓) ∈ V
6 simprl1 1216 . . . . . . 7 (((𝐹:𝐴𝐵𝐴𝑉) ∧ ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔))) → 𝑔:𝐴onto𝑃)
7 simp3 1136 . . . . . . . . 9 ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) → 𝑗:(𝐹𝐴)–1-1𝐵)
8 f1of1 6599 . . . . . . . . . 10 (𝑓:𝑃1-1-onto→(𝐹𝐴) → 𝑓:𝑃1-1→(𝐹𝐴))
983ad2ant2 1132 . . . . . . . . 9 ((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) → 𝑓:𝑃1-1→(𝐹𝐴))
10 f1co 6569 . . . . . . . . 9 ((𝑗:(𝐹𝐴)–1-1𝐵𝑓:𝑃1-1→(𝐹𝐴)) → (𝑗𝑓):𝑃1-1𝐵)
117, 9, 10syl2anc 588 . . . . . . . 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 1126 . . . . . 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 6553 . . . . . . . 8 ( = (𝑗𝑓) → (:𝑃1-1𝐵 ↔ (𝑗𝑓):𝑃1-1𝐵))
16 coeq1 5695 . . . . . . . . 9 ( = (𝑗𝑓) → (𝑔) = ((𝑗𝑓) ∘ 𝑔))
1716eqeq2d 2770 . . . . . . . 8 ( = (𝑗𝑓) → (𝐹 = (𝑔) ↔ 𝐹 = ((𝑗𝑓) ∘ 𝑔)))
1815, 173anbi23d 1437 . . . . . . 7 ( = (𝑗𝑓) → ((𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔)) ↔ (𝑔:𝐴onto𝑃 ∧ (𝑗𝑓):𝑃1-1𝐵𝐹 = ((𝑗𝑓) ∘ 𝑔))))
1918spcegv 3516 . . . . . 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 417 . . . 4 ((𝐹:𝐴𝐵𝐴𝑉) → (((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔)) → ∃(𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔))))
2221exlimdvv 1936 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → (∃𝑓𝑗((𝑔:𝐴onto𝑃𝑓:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑗:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑗𝑓) ∘ 𝑔)) → ∃(𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔))))
2322eximdv 1919 . 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 400   ∧ w3a 1085   = wceq 1539  ∃wex 1782   ∈ wcel 2112  {cab 2736  ∃wrex 3072  Vcvv 3410  {csn 4520  ◡ccnv 5521   “ cima 5525   ∘ ccom 5526  ⟶wf 6329  –1-1→wf1 6330  –onto→wfo 6331  –1-1-onto→wf1o 6332  ‘cfv 6333 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-op 4527  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341 This theorem is referenced by:  fundcmpsurinj  44284
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