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Theorem opabiota 6739
Description: Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 16-Nov-2013.)
Hypotheses
Ref Expression
opabiota.1 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
opabiota.2 (𝑥 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
opabiota (𝐵 ∈ dom 𝐹 → (𝐹𝐵) = (℩𝑦𝜓))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem opabiota
StepHypRef Expression
1 fveq2 6663 . . 3 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
2 opabiota.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
32iotabidv 6332 . . 3 (𝑥 = 𝐵 → (℩𝑦𝜑) = (℩𝑦𝜓))
41, 3eqeq12d 2834 . 2 (𝑥 = 𝐵 → ((𝐹𝑥) = (℩𝑦𝜑) ↔ (𝐹𝐵) = (℩𝑦𝜓)))
5 vex 3495 . . . 4 𝑥 ∈ V
65eldm 5762 . . 3 (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
7 nfiota1 6309 . . . . 5 𝑦(℩𝑦𝜑)
87nfeq2 2992 . . . 4 𝑦(𝐹𝑥) = (℩𝑦𝜑)
9 opabiota.1 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
109opabiotafun 6737 . . . . . 6 Fun 𝐹
11 funbrfv 6709 . . . . . 6 (Fun 𝐹 → (𝑥𝐹𝑦 → (𝐹𝑥) = 𝑦))
1210, 11ax-mp 5 . . . . 5 (𝑥𝐹𝑦 → (𝐹𝑥) = 𝑦)
13 df-br 5058 . . . . . . . 8 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
149eleq2i 2901 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}})
15 opabidw 5403 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} ↔ {𝑦𝜑} = {𝑦})
1613, 14, 153bitri 298 . . . . . . 7 (𝑥𝐹𝑦 ↔ {𝑦𝜑} = {𝑦})
17 vsnid 4592 . . . . . . . . 9 𝑦 ∈ {𝑦}
18 id 22 . . . . . . . . 9 ({𝑦𝜑} = {𝑦} → {𝑦𝜑} = {𝑦})
1917, 18eleqtrrid 2917 . . . . . . . 8 ({𝑦𝜑} = {𝑦} → 𝑦 ∈ {𝑦𝜑})
20 abid 2800 . . . . . . . 8 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
2119, 20sylib 219 . . . . . . 7 ({𝑦𝜑} = {𝑦} → 𝜑)
2216, 21sylbi 218 . . . . . 6 (𝑥𝐹𝑦𝜑)
23 vex 3495 . . . . . . . . 9 𝑦 ∈ V
245, 23breldm 5770 . . . . . . . 8 (𝑥𝐹𝑦𝑥 ∈ dom 𝐹)
259opabiotadm 6738 . . . . . . . . 9 dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
2625abeq2i 2945 . . . . . . . 8 (𝑥 ∈ dom 𝐹 ↔ ∃!𝑦𝜑)
2724, 26sylib 219 . . . . . . 7 (𝑥𝐹𝑦 → ∃!𝑦𝜑)
28 iota1 6325 . . . . . . 7 (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
2927, 28syl 17 . . . . . 6 (𝑥𝐹𝑦 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
3022, 29mpbid 233 . . . . 5 (𝑥𝐹𝑦 → (℩𝑦𝜑) = 𝑦)
3112, 30eqtr4d 2856 . . . 4 (𝑥𝐹𝑦 → (𝐹𝑥) = (℩𝑦𝜑))
328, 31exlimi 2207 . . 3 (∃𝑦 𝑥𝐹𝑦 → (𝐹𝑥) = (℩𝑦𝜑))
336, 32sylbi 218 . 2 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) = (℩𝑦𝜑))
344, 33vtoclga 3571 1 (𝐵 ∈ dom 𝐹 → (𝐹𝐵) = (℩𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wex 1771  wcel 2105  ∃!weu 2646  {cab 2796  {csn 4557  cop 4563   class class class wbr 5057  {copab 5119  dom cdm 5548  cio 6305  Fun wfun 6342  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by: (None)
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