Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfafv2 Structured version   Visualization version   GIF version

Theorem dfafv2 44296
Description: Alternative definition of (𝐹'''𝐴) using (𝐹𝐴) 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 6388 . . . . 5 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 simprr 773 . . . . . 6 ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ∃!𝑥 𝐴𝐹𝑥)
3 reuaiotaiota 44252 . . . . . 6 (∃!𝑥 𝐴𝐹𝑥 ↔ (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
42, 3sylib 221 . . . . 5 ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
51, 4syl5eq 2790 . . . 4 ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (𝐹𝐴) = (℩'𝑥𝐴𝐹𝑥))
6 eubrdm 44202 . . . . . . . . 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 44254 . . . . . 6 (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (℩'𝑥𝐴𝐹𝑥) = V)
119, 10sylib 221 . . . . 5 ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩'𝑥𝐴𝐹𝑥) = V)
1211eqcomd 2743 . . . 4 ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → V = (℩'𝑥𝐴𝐹𝑥))
135, 12ifeqda 4475 . . 3 (⊤ → if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹𝐴), V) = (℩'𝑥𝐴𝐹𝑥))
1413mptru 1550 . 2 if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹𝐴), V) = (℩'𝑥𝐴𝐹𝑥)
15 dfdfat2 44292 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
16 ifbi 4461 . . 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 44284 . 2 (𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥)
1914, 17, 183eqtr4ri 2776 1 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399   = wceq 1543  wtru 1544  wcel 2110  ∃!weu 2567  Vcvv 3408  ifcif 4439   class class class wbr 5053  dom cdm 5551  cio 6336  cfv 6380  ℩'caiota 44247   defAt wdfat 44280  '''cafv 44281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-res 5563  df-iota 6338  df-fun 6382  df-fv 6388  df-aiota 44249  df-dfat 44283  df-afv 44284
This theorem is referenced by:  afveq12d  44297  nfafv  44300  afvfundmfveq  44302  afvnfundmuv  44303  afvpcfv0  44310
  Copyright terms: Public domain W3C validator