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Theorem setrec2lem1 42211
Description: Lemma for setrec2 42213. 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 6205 . 2 (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎))
2 dmres 5417 . . . . . . 7 dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) = ({𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∩ dom 𝐹)
3 inss1 3831 . . . . . . 7 ({𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∩ dom 𝐹) ⊆ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
42, 3eqsstri 3633 . . . . . 6 dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ⊆ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
54sseli 3597 . . . . 5 (𝑎 ∈ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) → 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
65con3i 150 . . . 4 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ¬ 𝑎 ∈ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}))
7 ndmfv 6216 . . . 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 3201 . . . . . . 7 𝑎 ∈ V
10 breq1 4654 . . . . . . . 8 (𝑥 = 𝑎 → (𝑥𝐹𝑦𝑎𝐹𝑦))
1110eubidv 2489 . . . . . . 7 (𝑥 = 𝑎 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑎𝐹𝑦))
129, 11elab 3348 . . . . . 6 (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑎𝐹𝑦)
1312notbii 310 . . . . 5 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ¬ ∃!𝑦 𝑎𝐹𝑦)
1413biimpi 206 . . . 4 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ¬ ∃!𝑦 𝑎𝐹𝑦)
15 tz6.12-2 6180 . . . 4 (¬ ∃!𝑦 𝑎𝐹𝑦 → (𝐹𝑎) = ∅)
1614, 15syl 17 . . 3 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → (𝐹𝑎) = ∅)
178, 16eqtr4d 2658 . 2 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎))
181, 17pm2.61i 176 1 ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1482  wcel 1989  ∃!weu 2469  {cab 2607  cin 3571  c0 3913   class class class wbr 4651  dom cdm 5112  cres 5114  cfv 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-xp 5118  df-dm 5122  df-res 5124  df-iota 5849  df-fv 5894
This theorem is referenced by:  setrec2  42213
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