Theorem tz6.12-afv 47161

Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6846. (Contributed by Alexander van der Vekens, 29-Nov-2017.)

Assertion

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

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ V)
2		vex 3440	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3	2	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ V)
4		df-br 5093	. . . . . . . . . 10 ⊢ (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
5	4	biimpri 228	. . . . . . . . 9 ⊢ (⟨𝐴, 𝑦⟩ ∈ 𝐹 → 𝐴𝐹𝑦)
6	5	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴𝐹𝑦)
7		breldmg 5852	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑦 ∈ V ∧ 𝐴𝐹𝑦) → 𝐴 ∈ dom 𝐹)
8	1, 3, 6, 7	syl3anc 1373	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
9		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
10		velsn 4593	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11		breq1 5095	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 = 𝑥 → (𝐴𝐹𝑦 ↔ 𝑥𝐹𝑦))
12	4, 11	bitr3id 285	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 = 𝑥 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ 𝑥𝐹𝑦))
13	12	eqcoms 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ 𝑥𝐹𝑦))
14	13	eubidv 2579	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦 𝑥𝐹𝑦))
15	14	biimpd 229	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝑥𝐹𝑦))
16	10, 15	sylbi 217	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {𝐴} → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝑥𝐹𝑦))
17	16	com12 32	. . . . . . . . . . . 12 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ {𝐴} → ∃!𝑦 𝑥𝐹𝑦))
18	17	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} → ∃!𝑦 𝑥𝐹𝑦))
19	18	ralrimiv 3120	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
20		fnres 6609	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ {𝐴}) Fn {𝐴} ↔ ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
21		fnfun 6582	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ {𝐴}) Fn {𝐴} → Fun (𝐹 ↾ {𝐴}))
22	20, 21	sylbir 235	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦 → Fun (𝐹 ↾ {𝐴}))
23	19, 22	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → Fun (𝐹 ↾ {𝐴}))
24	9, 23	jca 511	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
25	24	ex 412	. . . . . . 7 ⊢ (𝐴 ∈ dom 𝐹 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
26	8, 25	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
27	26	impr 454	. . . . 5 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
28		df-dfat 47107	. . . . . 6 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
29		afvfundmfveq 47126	. . . . . 6 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴))
30	28, 29	sylbir 235	. . . . 5 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = (𝐹‘𝐴))
31	27, 30	syl 17	. . . 4 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹'''𝐴) = (𝐹‘𝐴))
32		tz6.12 6846	. . . . 5 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑦)
33	32	adantl 481	. . . 4 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹‘𝐴) = 𝑦)
34	31, 33	eqtrd 2764	. . 3 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹'''𝐴) = 𝑦)
35	34	ex 412	. 2 ⊢ (𝐴 ∈ V → ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦))
36		eu2ndop1stv 47113	. . . . 5 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → 𝐴 ∈ V)
37	36	pm2.24d 151	. . . 4 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → (¬ 𝐴 ∈ V → (𝐹'''𝐴) = 𝑦))
38	37	adantl 481	. . 3 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (¬ 𝐴 ∈ V → (𝐹'''𝐴) = 𝑦))
39	38	com12 32	. 2 ⊢ (¬ 𝐴 ∈ V → ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦))
40	35, 39	pm2.61i 182	1 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃!weu 2561  ∀wral 3044  Vcvv 3436  {csn 4577  ⟨cop 4583   class class class wbr 5092  dom cdm 5619   ↾ cres 5621  Fun wfun 6476   Fn wfn 6477  ‘cfv 6482   defAt wdfat 47104  '''cafv 47105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-res 5631  df-iota 6438  df-fun 6484  df-fn 6485  df-fv 6490  df-aiota 47073  df-dfat 47107  df-afv 47108

This theorem is referenced by:  tz6.12-1-afv  47162