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Theorem dfafv2 43703
 Description: Alternative definition of (𝐹'''𝐴) using (𝐹‘𝐴) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.) (Revised by AV, 25-Aug-2022.)
Assertion
Ref Expression
dfafv2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)

Proof of Theorem dfafv2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-fv 6332 . . . . 5 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 simprr 772 . . . . . 6 ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ∃!𝑥 𝐴𝐹𝑥)
3 reuaiotaiota 43660 . . . . . 6 (∃!𝑥 𝐴𝐹𝑥 ↔ (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
42, 3sylib 221 . . . . 5 ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
51, 4syl5eq 2845 . . . 4 ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (𝐹𝐴) = (℩'𝑥𝐴𝐹𝑥))
6 eubrdm 43643 . . . . . . . . 9 (∃!𝑥 𝐴𝐹𝑥𝐴 ∈ dom 𝐹)
76ancri 553 . . . . . . . 8 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
87con3i 157 . . . . . . 7 (¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → ¬ ∃!𝑥 𝐴𝐹𝑥)
98adantl 485 . . . . . 6 ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ¬ ∃!𝑥 𝐴𝐹𝑥)
10 aiotavb 43662 . . . . . 6 (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (℩'𝑥𝐴𝐹𝑥) = V)
119, 10sylib 221 . . . . 5 ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩'𝑥𝐴𝐹𝑥) = V)
1211eqcomd 2804 . . . 4 ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → V = (℩'𝑥𝐴𝐹𝑥))
135, 12ifeqda 4460 . . 3 (⊤ → if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹𝐴), V) = (℩'𝑥𝐴𝐹𝑥))
1413mptru 1545 . 2 if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹𝐴), V) = (℩'𝑥𝐴𝐹𝑥)
15 dfdfat2 43699 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
16 ifbi 4446 . . 3 ((𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → if(𝐹 defAt 𝐴, (𝐹𝐴), V) = if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹𝐴), V))
1715, 16ax-mp 5 . 2 if(𝐹 defAt 𝐴, (𝐹𝐴), V) = if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹𝐴), V)
18 df-afv 43691 . 2 (𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥)
1914, 17, 183eqtr4ri 2832 1 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  ∃!weu 2628  Vcvv 3441  ifcif 4425   class class class wbr 5030  dom cdm 5519  ℩cio 6281  ‘cfv 6324  ℩'caiota 43655   defAt wdfat 43687  '''cafv 43688 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 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  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-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  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 4801  df-int 4839  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-iota 6283  df-fun 6326  df-fv 6332  df-aiota 43657  df-dfat 43690  df-afv 43691 This theorem is referenced by:  afveq12d  43704  nfafv  43707  afvfundmfveq  43709  afvnfundmuv  43710  afvpcfv0  43717
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