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Theorem fnres 6695
Description: An equivalence for functionality of a restriction. Compare dffun8 6594. (Contributed by Mario Carneiro, 20-May-2015.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Assertion
Ref Expression
fnres ((𝐹𝐴) Fn 𝐴 ↔ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem fnres
StepHypRef Expression
1 ancom 460 . . 3 ((∀𝑥𝐴 ∃*𝑦 𝑥𝐹𝑦 ∧ ∀𝑥𝐴𝑦 𝑥𝐹𝑦) ↔ (∀𝑥𝐴𝑦 𝑥𝐹𝑦 ∧ ∀𝑥𝐴 ∃*𝑦 𝑥𝐹𝑦))
2 vex 3484 . . . . . . . . 9 𝑦 ∈ V
32brresi 6006 . . . . . . . 8 (𝑥(𝐹𝐴)𝑦 ↔ (𝑥𝐴𝑥𝐹𝑦))
43mobii 2548 . . . . . . 7 (∃*𝑦 𝑥(𝐹𝐴)𝑦 ↔ ∃*𝑦(𝑥𝐴𝑥𝐹𝑦))
5 moanimv 2619 . . . . . . 7 (∃*𝑦(𝑥𝐴𝑥𝐹𝑦) ↔ (𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
64, 5bitri 275 . . . . . 6 (∃*𝑦 𝑥(𝐹𝐴)𝑦 ↔ (𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
76albii 1819 . . . . 5 (∀𝑥∃*𝑦 𝑥(𝐹𝐴)𝑦 ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
8 relres 6023 . . . . . 6 Rel (𝐹𝐴)
9 dffun6 6574 . . . . . 6 (Fun (𝐹𝐴) ↔ (Rel (𝐹𝐴) ∧ ∀𝑥∃*𝑦 𝑥(𝐹𝐴)𝑦))
108, 9mpbiran 709 . . . . 5 (Fun (𝐹𝐴) ↔ ∀𝑥∃*𝑦 𝑥(𝐹𝐴)𝑦)
11 df-ral 3062 . . . . 5 (∀𝑥𝐴 ∃*𝑦 𝑥𝐹𝑦 ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
127, 10, 113bitr4i 303 . . . 4 (Fun (𝐹𝐴) ↔ ∀𝑥𝐴 ∃*𝑦 𝑥𝐹𝑦)
13 dmres 6030 . . . . . . 7 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
14 inss1 4237 . . . . . . 7 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
1513, 14eqsstri 4030 . . . . . 6 dom (𝐹𝐴) ⊆ 𝐴
16 eqss 3999 . . . . . 6 (dom (𝐹𝐴) = 𝐴 ↔ (dom (𝐹𝐴) ⊆ 𝐴𝐴 ⊆ dom (𝐹𝐴)))
1715, 16mpbiran 709 . . . . 5 (dom (𝐹𝐴) = 𝐴𝐴 ⊆ dom (𝐹𝐴))
18 dfss3 3972 . . . . . 6 (𝐴 ⊆ dom (𝐹𝐴) ↔ ∀𝑥𝐴 𝑥 ∈ dom (𝐹𝐴))
1913elin2 4203 . . . . . . . . 9 (𝑥 ∈ dom (𝐹𝐴) ↔ (𝑥𝐴𝑥 ∈ dom 𝐹))
2019baib 535 . . . . . . . 8 (𝑥𝐴 → (𝑥 ∈ dom (𝐹𝐴) ↔ 𝑥 ∈ dom 𝐹))
21 vex 3484 . . . . . . . . 9 𝑥 ∈ V
2221eldm 5911 . . . . . . . 8 (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
2320, 22bitrdi 287 . . . . . . 7 (𝑥𝐴 → (𝑥 ∈ dom (𝐹𝐴) ↔ ∃𝑦 𝑥𝐹𝑦))
2423ralbiia 3091 . . . . . 6 (∀𝑥𝐴 𝑥 ∈ dom (𝐹𝐴) ↔ ∀𝑥𝐴𝑦 𝑥𝐹𝑦)
2518, 24bitri 275 . . . . 5 (𝐴 ⊆ dom (𝐹𝐴) ↔ ∀𝑥𝐴𝑦 𝑥𝐹𝑦)
2617, 25bitri 275 . . . 4 (dom (𝐹𝐴) = 𝐴 ↔ ∀𝑥𝐴𝑦 𝑥𝐹𝑦)
2712, 26anbi12i 628 . . 3 ((Fun (𝐹𝐴) ∧ dom (𝐹𝐴) = 𝐴) ↔ (∀𝑥𝐴 ∃*𝑦 𝑥𝐹𝑦 ∧ ∀𝑥𝐴𝑦 𝑥𝐹𝑦))
28 r19.26 3111 . . 3 (∀𝑥𝐴 (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦) ↔ (∀𝑥𝐴𝑦 𝑥𝐹𝑦 ∧ ∀𝑥𝐴 ∃*𝑦 𝑥𝐹𝑦))
291, 27, 283bitr4i 303 . 2 ((Fun (𝐹𝐴) ∧ dom (𝐹𝐴) = 𝐴) ↔ ∀𝑥𝐴 (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
30 df-fn 6564 . 2 ((𝐹𝐴) Fn 𝐴 ↔ (Fun (𝐹𝐴) ∧ dom (𝐹𝐴) = 𝐴))
31 df-eu 2569 . . 3 (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
3231ralbii 3093 . 2 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥𝐴 (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
3329, 30, 323bitr4i 303 1 ((𝐹𝐴) Fn 𝐴 ↔ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  ∃*wmo 2538  ∃!weu 2568  wral 3061  cin 3950  wss 3951   class class class wbr 5143  dom cdm 5685  cres 5687  Rel wrel 5690  Fun wfun 6555   Fn wfn 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-fun 6563  df-fn 6564
This theorem is referenced by:  f1ompt  7131  omxpenlem  9113  tz6.12-afv  47185  tz6.12-afv2  47252
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