Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfdfat2 Structured version   Visualization version   GIF version

Theorem dfdfat2 40531
Description: Alternate definition of the predicate "defined at" not using the Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
Assertion
Ref Expression
dfdfat2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Proof of Theorem dfdfat2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-dfat 40516 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 relres 5387 . . . 4 Rel (𝐹 ↾ {𝐴})
3 dffun8 5877 . . . 4 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
42, 3mpbiran 952 . . 3 (Fun (𝐹 ↾ {𝐴}) ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
54anbi2i 729 . 2 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
6 vex 3189 . . . . . . . 8 𝑦 ∈ V
76brres 5364 . . . . . . 7 (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ {𝐴}))
87a1i 11 . . . . . 6 (𝐴 ∈ dom 𝐹 → (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ {𝐴})))
98eubidv 2489 . . . . 5 (𝐴 ∈ dom 𝐹 → (∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦(𝑥𝐹𝑦𝑥 ∈ {𝐴})))
109ralbidv 2980 . . . 4 (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥𝐹𝑦𝑥 ∈ {𝐴})))
11 eldmressnsn 5400 . . . . 5 (𝐴 ∈ dom 𝐹𝐴 ∈ dom (𝐹 ↾ {𝐴}))
12 eldmressn 40523 . . . . 5 (𝑥 ∈ dom (𝐹 ↾ {𝐴}) → 𝑥 = 𝐴)
13 breq1 4618 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
1413anbi1d 740 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝐹𝑦𝑥 ∈ {𝐴}) ↔ (𝐴𝐹𝑦𝑥 ∈ {𝐴})))
15 velsn 4166 . . . . . . . . 9 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
1615biimpri 218 . . . . . . . 8 (𝑥 = 𝐴𝑥 ∈ {𝐴})
1716biantrud 528 . . . . . . 7 (𝑥 = 𝐴 → (𝐴𝐹𝑦 ↔ (𝐴𝐹𝑦𝑥 ∈ {𝐴})))
1814, 17bitr4d 271 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝐹𝑦𝑥 ∈ {𝐴}) ↔ 𝐴𝐹𝑦))
1918eubidv 2489 . . . . 5 (𝑥 = 𝐴 → (∃!𝑦(𝑥𝐹𝑦𝑥 ∈ {𝐴}) ↔ ∃!𝑦 𝐴𝐹𝑦))
2011, 12, 19ralbinrald 40519 . . . 4 (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥𝐹𝑦𝑥 ∈ {𝐴}) ↔ ∃!𝑦 𝐴𝐹𝑦))
2110, 20bitrd 268 . . 3 (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦 𝐴𝐹𝑦))
2221pm5.32i 668 . 2 ((𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
231, 5, 223bitri 286 1 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  ∃!weu 2469  wral 2907  {csn 4150   class class class wbr 4615  dom cdm 5076  cres 5078  Rel wrel 5081  Fun wfun 5843   defAt wdfat 40513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-br 4616  df-opab 4676  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-res 5088  df-fun 5851  df-dfat 40516
This theorem is referenced by:  afveu  40553  rlimdmafv  40577
  Copyright terms: Public domain W3C validator