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Theorem setrec2lem1 42958
Description: Lemma for setrec2 42960. 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 6348 . 2 (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎))
2 dmres 5560 . . . . . . 7 dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) = ({𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∩ dom 𝐹)
3 inss1 3979 . . . . . . 7 ({𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∩ dom 𝐹) ⊆ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
42, 3eqsstri 3782 . . . . . 6 dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ⊆ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
54sseli 3746 . . . . 5 (𝑎 ∈ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) → 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
65con3i 151 . . . 4 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ¬ 𝑎 ∈ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}))
7 ndmfv 6359 . . . 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 3352 . . . . . . 7 𝑎 ∈ V
10 breq1 4787 . . . . . . . 8 (𝑥 = 𝑎 → (𝑥𝐹𝑦𝑎𝐹𝑦))
1110eubidv 2637 . . . . . . 7 (𝑥 = 𝑎 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑎𝐹𝑦))
129, 11elab 3499 . . . . . 6 (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑎𝐹𝑦)
1312notbii 309 . . . . 5 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ¬ ∃!𝑦 𝑎𝐹𝑦)
1413biimpi 206 . . . 4 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ¬ ∃!𝑦 𝑎𝐹𝑦)
15 tz6.12-2 6323 . . . 4 (¬ ∃!𝑦 𝑎𝐹𝑦 → (𝐹𝑎) = ∅)
1614, 15syl 17 . . 3 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → (𝐹𝑎) = ∅)
178, 16eqtr4d 2807 . 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 1630  wcel 2144  ∃!weu 2617  {cab 2756  cin 3720  c0 4061   class class class wbr 4784  dom cdm 5249  cres 5251  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-xp 5255  df-dm 5259  df-res 5261  df-iota 5994  df-fv 6039
This theorem is referenced by:  setrec2  42960
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