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Theorem tz6.12-afv2 45479
Description: Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6868. (Contributed by AV, 5-Sep-2022.)
Assertion
Ref Expression
tz6.12-afv2 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Proof of Theorem tz6.12-afv2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpl 484 . . . . . . . . 9 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ V)
2 vex 3450 . . . . . . . . . 10 𝑦 ∈ V
32a1i 11 . . . . . . . . 9 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ V)
4 df-br 5107 . . . . . . . . . . 11 (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
54biimpri 227 . . . . . . . . . 10 (⟨𝐴, 𝑦⟩ ∈ 𝐹𝐴𝐹𝑦)
65adantl 483 . . . . . . . . 9 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴𝐹𝑦)
7 breldmg 5866 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑦 ∈ V ∧ 𝐴𝐹𝑦) → 𝐴 ∈ dom 𝐹)
81, 3, 6, 7syl3anc 1372 . . . . . . . 8 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
9 simpl 484 . . . . . . . . . 10 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
10 velsn 4603 . . . . . . . . . . . . . . 15 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11 breq1 5109 . . . . . . . . . . . . . . . . . . 19 (𝐴 = 𝑥 → (𝐴𝐹𝑦𝑥𝐹𝑦))
124, 11bitr3id 285 . . . . . . . . . . . . . . . . . 18 (𝐴 = 𝑥 → (⟨𝐴, 𝑦⟩ ∈ 𝐹𝑥𝐹𝑦))
1312eqcoms 2745 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐴 → (⟨𝐴, 𝑦⟩ ∈ 𝐹𝑥𝐹𝑦))
1413eubidv 2585 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦 𝑥𝐹𝑦))
1514biimpd 228 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝑥𝐹𝑦))
1610, 15sylbi 216 . . . . . . . . . . . . . 14 (𝑥 ∈ {𝐴} → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝑥𝐹𝑦))
1716com12 32 . . . . . . . . . . . . 13 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ {𝐴} → ∃!𝑦 𝑥𝐹𝑦))
1817adantl 483 . . . . . . . . . . . 12 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} → ∃!𝑦 𝑥𝐹𝑦))
1918ralrimiv 3143 . . . . . . . . . . 11 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
20 fnres 6629 . . . . . . . . . . . 12 ((𝐹 ↾ {𝐴}) Fn {𝐴} ↔ ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
21 fnfun 6603 . . . . . . . . . . . 12 ((𝐹 ↾ {𝐴}) Fn {𝐴} → Fun (𝐹 ↾ {𝐴}))
2220, 21sylbir 234 . . . . . . . . . . 11 (∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦 → Fun (𝐹 ↾ {𝐴}))
2319, 22syl 17 . . . . . . . . . 10 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → Fun (𝐹 ↾ {𝐴}))
249, 23jca 513 . . . . . . . . 9 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2524ex 414 . . . . . . . 8 (𝐴 ∈ dom 𝐹 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
268, 25syl 17 . . . . . . 7 ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
2726impr 456 . . . . . 6 ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
28 df-dfat 45358 . . . . . 6 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2927, 28sylibr 233 . . . . 5 ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)) → 𝐹 defAt 𝐴)
30 dfatafv2iota 45449 . . . . 5 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑦𝐴𝐹𝑦))
3129, 30syl 17 . . . 4 ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹''''𝐴) = (℩𝑦𝐴𝐹𝑦))
324bicomi 223 . . . . . . . . 9 (⟨𝐴, 𝑦⟩ ∈ 𝐹𝐴𝐹𝑦)
3332eubii 2584 . . . . . . . 8 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦 𝐴𝐹𝑦)
3433biimpi 215 . . . . . . 7 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝐴𝐹𝑦)
355, 34anim12i 614 . . . . . 6 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦))
3635adantl 483 . . . . 5 ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦))
37 iota1 6474 . . . . . 6 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦))
3837biimpac 480 . . . . 5 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (℩𝑦𝐴𝐹𝑦) = 𝑦)
3936, 38syl 17 . . . 4 ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)) → (℩𝑦𝐴𝐹𝑦) = 𝑦)
4031, 39eqtrd 2777 . . 3 ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹''''𝐴) = 𝑦)
4140ex 414 . 2 (𝐴 ∈ V → ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦))
42 eu2ndop1stv 45364 . . . . 5 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹𝐴 ∈ V)
4342pm2.24d 151 . . . 4 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹 → (¬ 𝐴 ∈ V → (𝐹''''𝐴) = 𝑦))
4443adantl 483 . . 3 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (¬ 𝐴 ∈ V → (𝐹''''𝐴) = 𝑦))
4544com12 32 . 2 𝐴 ∈ V → ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦))
4641, 45pm2.61i 182 1 ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  ∃!weu 2567  wral 3065  Vcvv 3446  {csn 4587  cop 4593   class class class wbr 5106  dom cdm 5634  cres 5636  cio 6447  Fun wfun 6491   Fn wfn 6492   defAt wdfat 45355  ''''cafv2 45447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6449  df-fun 6499  df-fn 6500  df-dfat 45358  df-afv2 45448
This theorem is referenced by:  tz6.12-1-afv2  45480
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