Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tz6.12-afv Structured version   Visualization version   GIF version

Theorem tz6.12-afv 47190
Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6930. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
Assertion
Ref Expression
tz6.12-afv ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Proof of Theorem tz6.12-afv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . . . . . 8 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ V)
2 vex 3483 . . . . . . . . 9 𝑦 ∈ V
32a1i 11 . . . . . . . 8 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ V)
4 df-br 5143 . . . . . . . . . 10 (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
54biimpri 228 . . . . . . . . 9 (⟨𝐴, 𝑦⟩ ∈ 𝐹𝐴𝐹𝑦)
65adantl 481 . . . . . . . 8 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴𝐹𝑦)
7 breldmg 5919 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑦 ∈ V ∧ 𝐴𝐹𝑦) → 𝐴 ∈ dom 𝐹)
81, 3, 6, 7syl3anc 1372 . . . . . . 7 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
9 simpl 482 . . . . . . . . 9 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
10 velsn 4641 . . . . . . . . . . . . . 14 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11 breq1 5145 . . . . . . . . . . . . . . . . . 18 (𝐴 = 𝑥 → (𝐴𝐹𝑦𝑥𝐹𝑦))
124, 11bitr3id 285 . . . . . . . . . . . . . . . . 17 (𝐴 = 𝑥 → (⟨𝐴, 𝑦⟩ ∈ 𝐹𝑥𝐹𝑦))
1312eqcoms 2744 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴 → (⟨𝐴, 𝑦⟩ ∈ 𝐹𝑥𝐹𝑦))
1413eubidv 2585 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦 𝑥𝐹𝑦))
1514biimpd 229 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝑥𝐹𝑦))
1610, 15sylbi 217 . . . . . . . . . . . . 13 (𝑥 ∈ {𝐴} → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝑥𝐹𝑦))
1716com12 32 . . . . . . . . . . . 12 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ {𝐴} → ∃!𝑦 𝑥𝐹𝑦))
1817adantl 481 . . . . . . . . . . 11 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} → ∃!𝑦 𝑥𝐹𝑦))
1918ralrimiv 3144 . . . . . . . . . 10 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
20 fnres 6694 . . . . . . . . . . 11 ((𝐹 ↾ {𝐴}) Fn {𝐴} ↔ ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
21 fnfun 6667 . . . . . . . . . . 11 ((𝐹 ↾ {𝐴}) Fn {𝐴} → Fun (𝐹 ↾ {𝐴}))
2220, 21sylbir 235 . . . . . . . . . 10 (∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦 → Fun (𝐹 ↾ {𝐴}))
2319, 22syl 17 . . . . . . . . 9 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → Fun (𝐹 ↾ {𝐴}))
249, 23jca 511 . . . . . . . 8 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2524ex 412 . . . . . . 7 (𝐴 ∈ dom 𝐹 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
268, 25syl 17 . . . . . 6 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
2726impr 454 . . . . 5 ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
28 df-dfat 47136 . . . . . 6 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
29 afvfundmfveq 47155 . . . . . 6 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
3028, 29sylbir 235 . . . . 5 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = (𝐹𝐴))
3127, 30syl 17 . . . 4 ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹'''𝐴) = (𝐹𝐴))
32 tz6.12 6930 . . . . 5 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
3332adantl 481 . . . 4 ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹𝐴) = 𝑦)
3431, 33eqtrd 2776 . . 3 ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹'''𝐴) = 𝑦)
3534ex 412 . 2 (𝐴 ∈ V → ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦))
36 eu2ndop1stv 47142 . . . . 5 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹𝐴 ∈ V)
3736pm2.24d 151 . . . 4 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → (¬ 𝐴 ∈ V → (𝐹'''𝐴) = 𝑦))
3837adantl 481 . . 3 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (¬ 𝐴 ∈ V → (𝐹'''𝐴) = 𝑦))
3938com12 32 . 2 𝐴 ∈ V → ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦))
4035, 39pm2.61i 182 1 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  ∃!weu 2567  wral 3060  Vcvv 3479  {csn 4625  cop 4631   class class class wbr 5142  dom cdm 5684  cres 5686  Fun wfun 6554   Fn wfn 6555  cfv 6560   defAt wdfat 47133  '''cafv 47134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-aiota 47102  df-dfat 47136  df-afv 47137
This theorem is referenced by:  tz6.12-1-afv  47191
  Copyright terms: Public domain W3C validator