Theorem tz6.12-afv2 48035

Description: Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6909. (Contributed by AV, 5-Sep-2022.)

Assertion

Ref	Expression
tz6.12-afv2	⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 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.

Step	Hyp	Ref	Expression
1		simpl 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ V)
2		vex 3461	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
3	2	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ V)
4		df-br 5112	. . . . . . . . . . 11 ⊢ (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
5	4	biimpri 231	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝑦⟩ ∈ 𝐹 → 𝐴𝐹𝑦)
6	5	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴𝐹𝑦)
7		breldmg 5901	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑦 ∈ V ∧ 𝐴𝐹𝑦) → 𝐴 ∈ dom 𝐹)
8	1, 3, 6, 7	syl3anc 1398	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
9		simpl 488	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
10		velsn 4607	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11		breq1 5114	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 = 𝑥 → (𝐴𝐹𝑦 ↔ 𝑥𝐹𝑦))
12	4, 11	bitr3id 288	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 = 𝑥 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ 𝑥𝐹𝑦))
13	12	eqcoms 2773	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ 𝑥𝐹𝑦))
14	13	eubidv 2616	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦 𝑥𝐹𝑦))
15	14	biimpd 232	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝑥𝐹𝑦))
16	10, 15	sylbi 220	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {𝐴} → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝑥𝐹𝑦))
17	16	com12 33	. . . . . . . . . . . . 13 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ {𝐴} → ∃!𝑦 𝑥𝐹𝑦))
18	17	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} → ∃!𝑦 𝑥𝐹𝑦))
19	18	ralrimiv 3158	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
20		fnres 6666	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ {𝐴}) Fn {𝐴} ↔ ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
21		fnfun 6639	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ {𝐴}) Fn {𝐴} → Fun (𝐹 ↾ {𝐴}))
22	20, 21	sylbir 238	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦 → Fun (𝐹 ↾ {𝐴}))
23	19, 22	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → Fun (𝐹 ↾ {𝐴}))
24	9, 23	jca 521	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
25	24	ex 418	. . . . . . . 8 ⊢ (𝐴 ∈ dom 𝐹 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
26	8, 25	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
27	26	impr 460	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
28		df-dfat 47914	. . . . . 6 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
29	27, 28	sylibr 237	. . . . 5 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → 𝐹 defAt 𝐴)
30		dfatafv2iota 48005	. . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑦𝐴𝐹𝑦))
31	29, 30	syl 18	. . . 4 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹''''𝐴) = (℩𝑦𝐴𝐹𝑦))
32	4	bicomi 227	. . . . . . . . 9 ⊢ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ 𝐴𝐹𝑦)
33	32	eubii 2615	. . . . . . . 8 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦 𝐴𝐹𝑦)
34	33	biimpi 219	. . . . . . 7 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝐴𝐹𝑦)
35	5, 34	anim12i 625	. . . . . 6 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦))
36	35	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦))
37		iota1 6519	. . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦))
38	37	biimpac 484	. . . . 5 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (℩𝑦𝐴𝐹𝑦) = 𝑦)
39	36, 38	syl 18	. . . 4 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (℩𝑦𝐴𝐹𝑦) = 𝑦)
40	31, 39	eqtrd 2800	. . 3 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹''''𝐴) = 𝑦)
41	40	ex 418	. 2 ⊢ (𝐴 ∈ V → ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦))
42		eu2ndop1stv 47920	. . . . 5 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → 𝐴 ∈ V)
43	42	pm2.24d 152	. . . 4 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → (¬ 𝐴 ∈ V → (𝐹''''𝐴) = 𝑦))
44	43	adantl 487	. . 3 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (¬ 𝐴 ∈ V → (𝐹''''𝐴) = 𝑦))
45	44	com12 33	. 2 ⊢ (¬ 𝐴 ∈ V → ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦))
46	41, 45	pm2.61i 184	1 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃!weu 2598  ∀wral 3081  Vcvv 3457  {csn 4591  ⟨cop 4597   class class class wbr 5111  dom cdm 5663   ↾ cres 5665  ℩cio 6494  Fun wfun 6534   Fn wfn 6535   defAt wdfat 47911  ''''cafv2 48003

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-res 5675  df-iota 6496  df-fun 6542  df-fn 6543  df-dfat 47914  df-afv2 48004

This theorem is used by:  tz6.12-1-afv2  48036