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Theorem afv2eu 43837
 Description: The value of a function at a unique point, analogous to fveu 6637. (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 43670 . 2 (∃!𝑥 𝐴𝐹𝑥𝐴 ∈ V)
2 euex 2637 . . . . 5 (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
3 eldmg 5732 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
42, 3syl5ibrcom 250 . . . 4 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ V → 𝐴 ∈ dom 𝐹))
54impcom 411 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → 𝐴 ∈ dom 𝐹)
6 dfdfat2 43727 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
7 dfatafv2iota 43809 . . . . . . . . 9 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
8 iotauni 6300 . . . . . . . . 9 (∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = {𝑥𝐴𝐹𝑥})
97, 8sylan9eq 2853 . . . . . . . 8 ((𝐹 defAt 𝐴 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥})
109ex 416 . . . . . . 7 (𝐹 defAt 𝐴 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥}))
116, 10sylbir 238 . . . . . 6 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥}))
1211expcom 417 . . . . 5 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥})))
1312pm2.43a 54 . . . 4 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥}))
1413adantl 485 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥}))
155, 14mpd 15 . 2 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥})
161, 15mpancom 687 1 (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃!weu 2628  {cab 2776  Vcvv 3441  ∪ cuni 4801   class class class wbr 5031  dom cdm 5520  ℩cio 6282   defAt wdfat 43715  ''''cafv2 43807 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-res 5532  df-iota 6284  df-fun 6327  df-dfat 43718  df-afv2 43808 This theorem is referenced by: (None)
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