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Theorem afv2eu 47250
Description: The value of a function at a unique point, analogous to fveu 6895. (Contributed by AV, 5-Sep-2022.)
Assertion
Ref Expression
afv2eu (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem afv2eu
StepHypRef Expression
1 eubrv 47047 . 2 (∃!𝑥 𝐴𝐹𝑥𝐴 ∈ V)
2 euex 2577 . . . . 5 (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
3 eldmg 5909 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
42, 3syl5ibrcom 247 . . . 4 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ V → 𝐴 ∈ dom 𝐹))
54impcom 407 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → 𝐴 ∈ dom 𝐹)
6 dfdfat2 47140 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
7 dfatafv2iota 47222 . . . . . . . . 9 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
8 iotauni 6536 . . . . . . . . 9 (∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = {𝑥𝐴𝐹𝑥})
97, 8sylan9eq 2797 . . . . . . . 8 ((𝐹 defAt 𝐴 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥})
109ex 412 . . . . . . 7 (𝐹 defAt 𝐴 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥}))
116, 10sylbir 235 . . . . . 6 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥}))
1211expcom 413 . . . . 5 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥})))
1312pm2.43a 54 . . . 4 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥}))
1413adantl 481 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥}))
155, 14mpd 15 . 2 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥})
161, 15mpancom 688 1 (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  ∃!weu 2568  {cab 2714  Vcvv 3480   cuni 4907   class class class wbr 5143  dom cdm 5685  cio 6512   defAt wdfat 47128  ''''cafv2 47220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-dfat 47131  df-afv2 47221
This theorem is referenced by: (None)
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