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Theorem opabiota 6478
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 6404 . . 3 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
2 opabiota.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
32iotabidv 6081 . . 3 (𝑥 = 𝐵 → (℩𝑦𝜑) = (℩𝑦𝜓))
41, 3eqeq12d 2821 . 2 (𝑥 = 𝐵 → ((𝐹𝑥) = (℩𝑦𝜑) ↔ (𝐹𝐵) = (℩𝑦𝜓)))
5 vex 3394 . . . 4 𝑥 ∈ V
65eldm 5522 . . 3 (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
7 nfiota1 6062 . . . . 5 𝑦(℩𝑦𝜑)
87nfeq2 2964 . . . 4 𝑦(𝐹𝑥) = (℩𝑦𝜑)
9 opabiota.1 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
109opabiotafun 6476 . . . . . 6 Fun 𝐹
11 funbrfv 6450 . . . . . 6 (Fun 𝐹 → (𝑥𝐹𝑦 → (𝐹𝑥) = 𝑦))
1210, 11ax-mp 5 . . . . 5 (𝑥𝐹𝑦 → (𝐹𝑥) = 𝑦)
13 df-br 4845 . . . . . . . 8 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
149eleq2i 2877 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}})
15 opabid 5177 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} ↔ {𝑦𝜑} = {𝑦})
1613, 14, 153bitri 288 . . . . . . 7 (𝑥𝐹𝑦 ↔ {𝑦𝜑} = {𝑦})
17 vsnid 4403 . . . . . . . . 9 𝑦 ∈ {𝑦}
18 id 22 . . . . . . . . 9 ({𝑦𝜑} = {𝑦} → {𝑦𝜑} = {𝑦})
1917, 18syl5eleqr 2892 . . . . . . . 8 ({𝑦𝜑} = {𝑦} → 𝑦 ∈ {𝑦𝜑})
20 abid 2794 . . . . . . . 8 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
2119, 20sylib 209 . . . . . . 7 ({𝑦𝜑} = {𝑦} → 𝜑)
2216, 21sylbi 208 . . . . . 6 (𝑥𝐹𝑦𝜑)
23 vex 3394 . . . . . . . . 9 𝑦 ∈ V
245, 23breldm 5530 . . . . . . . 8 (𝑥𝐹𝑦𝑥 ∈ dom 𝐹)
259opabiotadm 6477 . . . . . . . . 9 dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
2625abeq2i 2919 . . . . . . . 8 (𝑥 ∈ dom 𝐹 ↔ ∃!𝑦𝜑)
2724, 26sylib 209 . . . . . . 7 (𝑥𝐹𝑦 → ∃!𝑦𝜑)
28 iota1 6074 . . . . . . 7 (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
2927, 28syl 17 . . . . . 6 (𝑥𝐹𝑦 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
3022, 29mpbid 223 . . . . 5 (𝑥𝐹𝑦 → (℩𝑦𝜑) = 𝑦)
3112, 30eqtr4d 2843 . . . 4 (𝑥𝐹𝑦 → (𝐹𝑥) = (℩𝑦𝜑))
328, 31exlimi 2253 . . 3 (∃𝑦 𝑥𝐹𝑦 → (𝐹𝑥) = (℩𝑦𝜑))
336, 32sylbi 208 . 2 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) = (℩𝑦𝜑))
344, 33vtoclga 3465 1 (𝐵 ∈ dom 𝐹 → (𝐹𝐵) = (℩𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1637  wex 1859  wcel 2156  ∃!weu 2630  {cab 2792  {csn 4370  cop 4376   class class class wbr 4844  {copab 4906  dom cdm 5311  cio 6058  Fun wfun 6091  cfv 6097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6060  df-fun 6099  df-fv 6105
This theorem is referenced by: (None)
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