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Theorem fundcmpsurinjALT 46080
Description: Alternate proof of fundcmpsurinj 46077, based on fundcmpsurinjimaid 46079: Every function 𝐹:𝐴𝐵 can be decomposed into a surjective and an injective function. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by AV, 13-Mar-2024.)
Assertion
Ref Expression
fundcmpsurinjALT ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑝(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)))
Distinct variable groups:   𝐴,𝑔,,𝑝   𝐵,𝑔,,𝑝   𝑔,𝐹,,𝑝
Allowed substitution hints:   𝑉(𝑔,,𝑝)

Proof of Theorem fundcmpsurinjALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mptexg 7223 . . . 4 (𝐴𝑉 → (𝑦𝐴 ↦ (𝐹𝑦)) ∈ V)
21adantl 483 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → (𝑦𝐴 ↦ (𝐹𝑦)) ∈ V)
3 ffun 6721 . . . . 5 (𝐹:𝐴𝐵 → Fun 𝐹)
4 funimaexg 6635 . . . . 5 ((Fun 𝐹𝐴𝑉) → (𝐹𝐴) ∈ V)
53, 4sylan 581 . . . 4 ((𝐹:𝐴𝐵𝐴𝑉) → (𝐹𝐴) ∈ V)
65resiexd 7218 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → ( I ↾ (𝐹𝐴)) ∈ V)
72, 6, 53jca 1129 . 2 ((𝐹:𝐴𝐵𝐴𝑉) → ((𝑦𝐴 ↦ (𝐹𝑦)) ∈ V ∧ ( I ↾ (𝐹𝐴)) ∈ V ∧ (𝐹𝐴) ∈ V))
8 eqid 2733 . . . 4 (𝐹𝐴) = (𝐹𝐴)
9 eqid 2733 . . . 4 (𝑦𝐴 ↦ (𝐹𝑦)) = (𝑦𝐴 ↦ (𝐹𝑦))
10 eqid 2733 . . . 4 ( I ↾ (𝐹𝐴)) = ( I ↾ (𝐹𝐴))
118, 9, 10fundcmpsurinjimaid 46079 . . 3 (𝐹:𝐴𝐵 → ((𝑦𝐴 ↦ (𝐹𝑦)):𝐴onto→(𝐹𝐴) ∧ ( I ↾ (𝐹𝐴)):(𝐹𝐴)–1-1𝐵𝐹 = (( I ↾ (𝐹𝐴)) ∘ (𝑦𝐴 ↦ (𝐹𝑦)))))
1211adantr 482 . 2 ((𝐹:𝐴𝐵𝐴𝑉) → ((𝑦𝐴 ↦ (𝐹𝑦)):𝐴onto→(𝐹𝐴) ∧ ( I ↾ (𝐹𝐴)):(𝐹𝐴)–1-1𝐵𝐹 = (( I ↾ (𝐹𝐴)) ∘ (𝑦𝐴 ↦ (𝐹𝑦)))))
13 simp1 1137 . . . . 5 ((𝑔 = (𝑦𝐴 ↦ (𝐹𝑦)) ∧ = ( I ↾ (𝐹𝐴)) ∧ 𝑝 = (𝐹𝐴)) → 𝑔 = (𝑦𝐴 ↦ (𝐹𝑦)))
14 eqidd 2734 . . . . 5 ((𝑔 = (𝑦𝐴 ↦ (𝐹𝑦)) ∧ = ( I ↾ (𝐹𝐴)) ∧ 𝑝 = (𝐹𝐴)) → 𝐴 = 𝐴)
15 simp3 1139 . . . . 5 ((𝑔 = (𝑦𝐴 ↦ (𝐹𝑦)) ∧ = ( I ↾ (𝐹𝐴)) ∧ 𝑝 = (𝐹𝐴)) → 𝑝 = (𝐹𝐴))
1613, 14, 15foeq123d 6827 . . . 4 ((𝑔 = (𝑦𝐴 ↦ (𝐹𝑦)) ∧ = ( I ↾ (𝐹𝐴)) ∧ 𝑝 = (𝐹𝐴)) → (𝑔:𝐴onto𝑝 ↔ (𝑦𝐴 ↦ (𝐹𝑦)):𝐴onto→(𝐹𝐴)))
17 simpl 484 . . . . . 6 (( = ( I ↾ (𝐹𝐴)) ∧ 𝑝 = (𝐹𝐴)) → = ( I ↾ (𝐹𝐴)))
18 simpr 486 . . . . . 6 (( = ( I ↾ (𝐹𝐴)) ∧ 𝑝 = (𝐹𝐴)) → 𝑝 = (𝐹𝐴))
19 eqidd 2734 . . . . . 6 (( = ( I ↾ (𝐹𝐴)) ∧ 𝑝 = (𝐹𝐴)) → 𝐵 = 𝐵)
2017, 18, 19f1eq123d 6826 . . . . 5 (( = ( I ↾ (𝐹𝐴)) ∧ 𝑝 = (𝐹𝐴)) → (:𝑝1-1𝐵 ↔ ( I ↾ (𝐹𝐴)):(𝐹𝐴)–1-1𝐵))
21203adant1 1131 . . . 4 ((𝑔 = (𝑦𝐴 ↦ (𝐹𝑦)) ∧ = ( I ↾ (𝐹𝐴)) ∧ 𝑝 = (𝐹𝐴)) → (:𝑝1-1𝐵 ↔ ( I ↾ (𝐹𝐴)):(𝐹𝐴)–1-1𝐵))
22 simpl 484 . . . . . . . 8 (( = ( I ↾ (𝐹𝐴)) ∧ 𝑔 = (𝑦𝐴 ↦ (𝐹𝑦))) → = ( I ↾ (𝐹𝐴)))
23 simpr 486 . . . . . . . 8 (( = ( I ↾ (𝐹𝐴)) ∧ 𝑔 = (𝑦𝐴 ↦ (𝐹𝑦))) → 𝑔 = (𝑦𝐴 ↦ (𝐹𝑦)))
2422, 23coeq12d 5865 . . . . . . 7 (( = ( I ↾ (𝐹𝐴)) ∧ 𝑔 = (𝑦𝐴 ↦ (𝐹𝑦))) → (𝑔) = (( I ↾ (𝐹𝐴)) ∘ (𝑦𝐴 ↦ (𝐹𝑦))))
2524ancoms 460 . . . . . 6 ((𝑔 = (𝑦𝐴 ↦ (𝐹𝑦)) ∧ = ( I ↾ (𝐹𝐴))) → (𝑔) = (( I ↾ (𝐹𝐴)) ∘ (𝑦𝐴 ↦ (𝐹𝑦))))
26253adant3 1133 . . . . 5 ((𝑔 = (𝑦𝐴 ↦ (𝐹𝑦)) ∧ = ( I ↾ (𝐹𝐴)) ∧ 𝑝 = (𝐹𝐴)) → (𝑔) = (( I ↾ (𝐹𝐴)) ∘ (𝑦𝐴 ↦ (𝐹𝑦))))
2726eqeq2d 2744 . . . 4 ((𝑔 = (𝑦𝐴 ↦ (𝐹𝑦)) ∧ = ( I ↾ (𝐹𝐴)) ∧ 𝑝 = (𝐹𝐴)) → (𝐹 = (𝑔) ↔ 𝐹 = (( I ↾ (𝐹𝐴)) ∘ (𝑦𝐴 ↦ (𝐹𝑦)))))
2816, 21, 273anbi123d 1437 . . 3 ((𝑔 = (𝑦𝐴 ↦ (𝐹𝑦)) ∧ = ( I ↾ (𝐹𝐴)) ∧ 𝑝 = (𝐹𝐴)) → ((𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)) ↔ ((𝑦𝐴 ↦ (𝐹𝑦)):𝐴onto→(𝐹𝐴) ∧ ( I ↾ (𝐹𝐴)):(𝐹𝐴)–1-1𝐵𝐹 = (( I ↾ (𝐹𝐴)) ∘ (𝑦𝐴 ↦ (𝐹𝑦))))))
2928spc3egv 3594 . 2 (((𝑦𝐴 ↦ (𝐹𝑦)) ∈ V ∧ ( I ↾ (𝐹𝐴)) ∈ V ∧ (𝐹𝐴) ∈ V) → (((𝑦𝐴 ↦ (𝐹𝑦)):𝐴onto→(𝐹𝐴) ∧ ( I ↾ (𝐹𝐴)):(𝐹𝐴)–1-1𝐵𝐹 = (( I ↾ (𝐹𝐴)) ∘ (𝑦𝐴 ↦ (𝐹𝑦)))) → ∃𝑔𝑝(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔))))
307, 12, 29sylc 65 1 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑝(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  Vcvv 3475  cmpt 5232   I cid 5574  cres 5679  cima 5680  ccom 5681  Fun wfun 6538  wf 6540  1-1wf1 6541  ontowfo 6542  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-pr 5428
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-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-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: (None)
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