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 46484
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 6550 . . . . 5 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 simprr 772 . . . . . 6 ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ∃!𝑥 𝐴𝐹𝑥)
3 reuaiotaiota 46440 . . . . . 6 (∃!𝑥 𝐴𝐹𝑥 ↔ (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
42, 3sylib 217 . . . . 5 ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
51, 4eqtrid 2779 . . . 4 ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (𝐹𝐴) = (℩'𝑥𝐴𝐹𝑥))
6 eubrdm 46390 . . . . . . . . 9 (∃!𝑥 𝐴𝐹𝑥𝐴 ∈ dom 𝐹)
76ancri 549 . . . . . . . 8 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
87con3i 154 . . . . . . 7 (¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → ¬ ∃!𝑥 𝐴𝐹𝑥)
98adantl 481 . . . . . 6 ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ¬ ∃!𝑥 𝐴𝐹𝑥)
10 aiotavb 46442 . . . . . 6 (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (℩'𝑥𝐴𝐹𝑥) = V)
119, 10sylib 217 . . . . 5 ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩'𝑥𝐴𝐹𝑥) = V)
1211eqcomd 2733 . . . 4 ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → V = (℩'𝑥𝐴𝐹𝑥))
135, 12ifeqda 4560 . . 3 (⊤ → if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹𝐴), V) = (℩'𝑥𝐴𝐹𝑥))
1413mptru 1541 . 2 if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹𝐴), V) = (℩'𝑥𝐴𝐹𝑥)
15 dfdfat2 46480 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
16 ifbi 4546 . . 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 46472 . 2 (𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥)
1914, 17, 183eqtr4ri 2766 1 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395   = wceq 1534  wtru 1535  wcel 2099  ∃!weu 2557  Vcvv 3469  ifcif 4524   class class class wbr 5142  dom cdm 5672  cio 6492  cfv 6542  ℩'caiota 46435   defAt wdfat 46468  '''cafv 46469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-iota 6494  df-fun 6544  df-fv 6550  df-aiota 46437  df-dfat 46471  df-afv 46472
This theorem is referenced by:  afveq12d  46485  nfafv  46488  afvfundmfveq  46490  afvnfundmuv  46491  afvpcfv0  46498
  Copyright terms: Public domain W3C validator