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Theorem setrec2lem1 42239
 Description: Lemma for setrec2 42241. The functional part of 𝐹 has the same values as 𝐹. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.)
Assertion
Ref Expression
setrec2lem1 ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎)
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑎,𝑦
Allowed substitution hint:   𝐹(𝑎)

Proof of Theorem setrec2lem1
StepHypRef Expression
1 fvres 6117 . 2 (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎))
2 dmres 5339 . . . . . . 7 dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) = ({𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∩ dom 𝐹)
3 inss1 3795 . . . . . . 7 ({𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∩ dom 𝐹) ⊆ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
42, 3eqsstri 3598 . . . . . 6 dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ⊆ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
54sseli 3564 . . . . 5 (𝑎 ∈ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) → 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
65con3i 149 . . . 4 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ¬ 𝑎 ∈ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}))
7 ndmfv 6128 . . . 4 𝑎 ∈ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = ∅)
86, 7syl 17 . . 3 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = ∅)
9 vex 3176 . . . . . . 7 𝑎 ∈ V
10 breq1 4586 . . . . . . . 8 (𝑥 = 𝑎 → (𝑥𝐹𝑦𝑎𝐹𝑦))
1110eubidv 2478 . . . . . . 7 (𝑥 = 𝑎 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑎𝐹𝑦))
129, 11elab 3319 . . . . . 6 (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑎𝐹𝑦)
1312notbii 309 . . . . 5 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ¬ ∃!𝑦 𝑎𝐹𝑦)
1413biimpi 205 . . . 4 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ¬ ∃!𝑦 𝑎𝐹𝑦)
15 tz6.12-2 6094 . . . 4 (¬ ∃!𝑦 𝑎𝐹𝑦 → (𝐹𝑎) = ∅)
1614, 15syl 17 . . 3 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → (𝐹𝑎) = ∅)
178, 16eqtr4d 2647 . 2 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎))
181, 17pm2.61i 175 1 ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  ∃!weu 2458  {cab 2596   ∩ cin 3539  ∅c0 3874   class class class wbr 4583  dom cdm 5038   ↾ cres 5040  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-res 5050  df-iota 5768  df-fv 5812 This theorem is referenced by:  setrec2  42241
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