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 41780
Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6441. (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 470 . . . . . . . 8 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ V)
2 vex 3405 . . . . . . . . 9 𝑦 ∈ V
32a1i 11 . . . . . . . 8 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ V)
4 df-br 4856 . . . . . . . . . 10 (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
54biimpri 219 . . . . . . . . 9 (⟨𝐴, 𝑦⟩ ∈ 𝐹𝐴𝐹𝑦)
65adantl 469 . . . . . . . 8 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴𝐹𝑦)
7 breldmg 5545 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑦 ∈ V ∧ 𝐴𝐹𝑦) → 𝐴 ∈ dom 𝐹)
81, 3, 6, 7syl3anc 1483 . . . . . . 7 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
9 simpl 470 . . . . . . . . 9 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
10 velsn 4397 . . . . . . . . . . . . . 14 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11 breq1 4858 . . . . . . . . . . . . . . . . . 18 (𝐴 = 𝑥 → (𝐴𝐹𝑦𝑥𝐹𝑦))
124, 11syl5bbr 276 . . . . . . . . . . . . . . . . 17 (𝐴 = 𝑥 → (⟨𝐴, 𝑦⟩ ∈ 𝐹𝑥𝐹𝑦))
1312eqcoms 2825 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴 → (⟨𝐴, 𝑦⟩ ∈ 𝐹𝑥𝐹𝑦))
1413eubidv 2647 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦 𝑥𝐹𝑦))
1514biimpd 220 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝑥𝐹𝑦))
1610, 15sylbi 208 . . . . . . . . . . . . 13 (𝑥 ∈ {𝐴} → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝑥𝐹𝑦))
1716com12 32 . . . . . . . . . . . 12 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ {𝐴} → ∃!𝑦 𝑥𝐹𝑦))
1817adantl 469 . . . . . . . . . . 11 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} → ∃!𝑦 𝑥𝐹𝑦))
1918ralrimiv 3164 . . . . . . . . . 10 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
20 fnres 6228 . . . . . . . . . . 11 ((𝐹 ↾ {𝐴}) Fn {𝐴} ↔ ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
21 fnfun 6209 . . . . . . . . . . 11 ((𝐹 ↾ {𝐴}) Fn {𝐴} → Fun (𝐹 ↾ {𝐴}))
2220, 21sylbir 226 . . . . . . . . . 10 (∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦 → Fun (𝐹 ↾ {𝐴}))
2319, 22syl 17 . . . . . . . . 9 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → Fun (𝐹 ↾ {𝐴}))
249, 23jca 503 . . . . . . . 8 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2524ex 399 . . . . . . 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 444 . . . . 5 ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
28 df-dfat 41726 . . . . . 6 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
29 afvfundmfveq 41745 . . . . . 6 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
3028, 29sylbir 226 . . . . 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 6441 . . . . 5 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
3332adantl 469 . . . 4 ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹𝐴) = 𝑦)
3431, 33eqtrd 2851 . . 3 ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹'''𝐴) = 𝑦)
3534ex 399 . 2 (𝐴 ∈ V → ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦))
36 eu2ndop1stv 41732 . . . . 5 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹𝐴 ∈ V)
3736pm2.24d 148 . . . 4 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → (¬ 𝐴 ∈ V → (𝐹'''𝐴) = 𝑦))
3837adantl 469 . . 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 176 1 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  ∃!weu 2641  wral 3107  Vcvv 3402  {csn 4381  cop 4387   class class class wbr 4855  dom cdm 5324  cres 5326  Fun wfun 6105   Fn wfn 6106  cfv 6111   defAt wdfat 41723  '''cafv 41724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-int 4681  df-br 4856  df-opab 4918  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-res 5336  df-iota 6074  df-fun 6113  df-fn 6114  df-fv 6119  df-aiota 41687  df-dfat 41726  df-afv 41727
This theorem is referenced by:  tz6.12-1-afv  41781
  Copyright terms: Public domain W3C validator