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Theorem fundcmpsurinj 46562
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 7940 . . . 4 (𝐴𝑉 → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∈ V)
21adantl 481 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∈ V)
3 fveq2 6881 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
43sneqd 4632 . . . . . . . 8 (𝑥 = 𝑦 → {(𝐹𝑥)} = {(𝐹𝑦)})
54imaeq2d 6049 . . . . . . 7 (𝑥 = 𝑦 → (𝐹 “ {(𝐹𝑥)}) = (𝐹 “ {(𝐹𝑦)}))
65eqeq2d 2735 . . . . . 6 (𝑥 = 𝑦 → (𝑧 = (𝐹 “ {(𝐹𝑥)}) ↔ 𝑧 = (𝐹 “ {(𝐹𝑦)})))
76cbvrexvw 3227 . . . . 5 (∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)}) ↔ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)}))
87abbii 2794 . . . 4 {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}
98fundcmpsurinjpreimafv 46561 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔(𝑔:𝐴onto→{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∧ :{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}–1-1𝐵𝐹 = (𝑔)))
10 foeq3 6793 . . . . 5 (𝑝 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} → (𝑔:𝐴onto𝑝𝑔:𝐴onto→{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}))
11 f1eq2 6773 . . . . 5 (𝑝 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} → (:𝑝1-1𝐵:{𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}–1-1𝐵))
1210, 113anbi12d 1433 . . . 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 1919 . . 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 3580 . 2 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑝𝑔(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)))
15 exrot3 2157 . 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 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2701  wrex 3062  Vcvv 3466  {csn 4620  ccnv 5665  cima 5669  ccom 5670  wf 6529  1-1wf1 6530  ontowfo 6531  cfv 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541
This theorem is referenced by: (None)
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