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Theorem opabiota 6300
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 6229 . . 3 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
2 opabiota.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
32iotabidv 5910 . . 3 (𝑥 = 𝐵 → (℩𝑦𝜑) = (℩𝑦𝜓))
41, 3eqeq12d 2666 . 2 (𝑥 = 𝐵 → ((𝐹𝑥) = (℩𝑦𝜑) ↔ (𝐹𝐵) = (℩𝑦𝜓)))
5 vex 3234 . . . 4 𝑥 ∈ V
65eldm 5353 . . 3 (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
7 nfiota1 5891 . . . . 5 𝑦(℩𝑦𝜑)
87nfeq2 2809 . . . 4 𝑦(𝐹𝑥) = (℩𝑦𝜑)
9 opabiota.1 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
109opabiotafun 6298 . . . . . 6 Fun 𝐹
11 funbrfv 6272 . . . . . 6 (Fun 𝐹 → (𝑥𝐹𝑦 → (𝐹𝑥) = 𝑦))
1210, 11ax-mp 5 . . . . 5 (𝑥𝐹𝑦 → (𝐹𝑥) = 𝑦)
13 df-br 4686 . . . . . . . 8 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
149eleq2i 2722 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}})
15 opabid 5011 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} ↔ {𝑦𝜑} = {𝑦})
1613, 14, 153bitri 286 . . . . . . 7 (𝑥𝐹𝑦 ↔ {𝑦𝜑} = {𝑦})
17 vsnid 4242 . . . . . . . . 9 𝑦 ∈ {𝑦}
18 id 22 . . . . . . . . 9 ({𝑦𝜑} = {𝑦} → {𝑦𝜑} = {𝑦})
1917, 18syl5eleqr 2737 . . . . . . . 8 ({𝑦𝜑} = {𝑦} → 𝑦 ∈ {𝑦𝜑})
20 abid 2639 . . . . . . . 8 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
2119, 20sylib 208 . . . . . . 7 ({𝑦𝜑} = {𝑦} → 𝜑)
2216, 21sylbi 207 . . . . . 6 (𝑥𝐹𝑦𝜑)
23 vex 3234 . . . . . . . . 9 𝑦 ∈ V
245, 23breldm 5361 . . . . . . . 8 (𝑥𝐹𝑦𝑥 ∈ dom 𝐹)
259opabiotadm 6299 . . . . . . . . 9 dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
2625abeq2i 2764 . . . . . . . 8 (𝑥 ∈ dom 𝐹 ↔ ∃!𝑦𝜑)
2724, 26sylib 208 . . . . . . 7 (𝑥𝐹𝑦 → ∃!𝑦𝜑)
28 iota1 5903 . . . . . . 7 (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
2927, 28syl 17 . . . . . 6 (𝑥𝐹𝑦 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
3022, 29mpbid 222 . . . . 5 (𝑥𝐹𝑦 → (℩𝑦𝜑) = 𝑦)
3112, 30eqtr4d 2688 . . . 4 (𝑥𝐹𝑦 → (𝐹𝑥) = (℩𝑦𝜑))
328, 31exlimi 2124 . . 3 (∃𝑦 𝑥𝐹𝑦 → (𝐹𝑥) = (℩𝑦𝜑))
336, 32sylbi 207 . 2 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) = (℩𝑦𝜑))
344, 33vtoclga 3303 1 (𝐵 ∈ dom 𝐹 → (𝐹𝐵) = (℩𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wex 1744  wcel 2030  ∃!weu 2498  {cab 2637  {csn 4210  cop 4216   class class class wbr 4685  {copab 4745  dom cdm 5143  cio 5887  Fun wfun 5920  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by: (None)
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