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Theorem fundcmpsurinj 44861
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 7803 . . . 4 (𝐴𝑉 → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∈ V)
21adantl 482 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∈ V)
3 fveq2 6774 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
43sneqd 4573 . . . . . . . 8 (𝑥 = 𝑦 → {(𝐹𝑥)} = {(𝐹𝑦)})
54imaeq2d 5969 . . . . . . 7 (𝑥 = 𝑦 → (𝐹 “ {(𝐹𝑥)}) = (𝐹 “ {(𝐹𝑦)}))
65eqeq2d 2749 . . . . . 6 (𝑥 = 𝑦 → (𝑧 = (𝐹 “ {(𝐹𝑥)}) ↔ 𝑧 = (𝐹 “ {(𝐹𝑦)})))
76cbvrexvw 3384 . . . . 5 (∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)}) ↔ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)}))
87abbii 2808 . . . 4 {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}
98fundcmpsurinjpreimafv 44860 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔(𝑔:𝐴onto→{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∧ :{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}–1-1𝐵𝐹 = (𝑔)))
10 foeq3 6686 . . . . 5 (𝑝 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} → (𝑔:𝐴onto𝑝𝑔:𝐴onto→{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}))
11 f1eq2 6666 . . . . 5 (𝑝 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} → (:𝑝1-1𝐵:{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}–1-1𝐵))
1210, 113anbi12d 1436 . . . 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 1927 . . 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 3537 . 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 217 1 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑝(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wrex 3065  Vcvv 3432  {csn 4561  ccnv 5588  cima 5592  ccom 5593  wf 6429  1-1wf1 6430  ontowfo 6431  cfv 6433
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441
This theorem is referenced by: (None)
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