Theorem tz6.12-afv2 47236

Description: Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6906. (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 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ V)
2		vex 3468	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
3	2	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ V)
4		df-br 5125	. . . . . . . . . . 11 ⊢ (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
5	4	biimpri 228	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝑦⟩ ∈ 𝐹 → 𝐴𝐹𝑦)
6	5	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴𝐹𝑦)
7		breldmg 5894	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑦 ∈ V ∧ 𝐴𝐹𝑦) → 𝐴 ∈ dom 𝐹)
8	1, 3, 6, 7	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
9		simpl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
10		velsn 4622	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11		breq1 5127	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 = 𝑥 → (𝐴𝐹𝑦 ↔ 𝑥𝐹𝑦))
12	4, 11	bitr3id 285	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 = 𝑥 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ 𝑥𝐹𝑦))
13	12	eqcoms 2744	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ 𝑥𝐹𝑦))
14	13	eubidv 2586	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦 𝑥𝐹𝑦))
15	14	biimpd 229	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝑥𝐹𝑦))
16	10, 15	sylbi 217	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {𝐴} → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝑥𝐹𝑦))
17	16	com12 32	. . . . . . . . . . . . 13 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ {𝐴} → ∃!𝑦 𝑥𝐹𝑦))
18	17	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} → ∃!𝑦 𝑥𝐹𝑦))
19	18	ralrimiv 3132	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
20		fnres 6670	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ {𝐴}) Fn {𝐴} ↔ ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
21		fnfun 6643	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ {𝐴}) Fn {𝐴} → Fun (𝐹 ↾ {𝐴}))
22	20, 21	sylbir 235	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦 → Fun (𝐹 ↾ {𝐴}))
23	19, 22	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → Fun (𝐹 ↾ {𝐴}))
24	9, 23	jca 511	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
25	24	ex 412	. . . . . . . 8 ⊢ (𝐴 ∈ dom 𝐹 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
26	8, 25	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
27	26	impr 454	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
28		df-dfat 47115	. . . . . 6 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
29	27, 28	sylibr 234	. . . . 5 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → 𝐹 defAt 𝐴)
30		dfatafv2iota 47206	. . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑦𝐴𝐹𝑦))
31	29, 30	syl 17	. . . 4 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹''''𝐴) = (℩𝑦𝐴𝐹𝑦))
32	4	bicomi 224	. . . . . . . . 9 ⊢ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ 𝐴𝐹𝑦)
33	32	eubii 2585	. . . . . . . 8 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦 𝐴𝐹𝑦)
34	33	biimpi 216	. . . . . . 7 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝐴𝐹𝑦)
35	5, 34	anim12i 613	. . . . . 6 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦))
36	35	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦))
37		iota1 6513	. . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦))
38	37	biimpac 478	. . . . 5 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (℩𝑦𝐴𝐹𝑦) = 𝑦)
39	36, 38	syl 17	. . . 4 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (℩𝑦𝐴𝐹𝑦) = 𝑦)
40	31, 39	eqtrd 2771	. . 3 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹''''𝐴) = 𝑦)
41	40	ex 412	. 2 ⊢ (𝐴 ∈ V → ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦))
42		eu2ndop1stv 47121	. . . . 5 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → 𝐴 ∈ V)
43	42	pm2.24d 151	. . . 4 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → (¬ 𝐴 ∈ V → (𝐹''''𝐴) = 𝑦))
44	43	adantl 481	. . 3 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (¬ 𝐴 ∈ V → (𝐹''''𝐴) = 𝑦))
45	44	com12 32	. 2 ⊢ (¬ 𝐴 ∈ V → ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦))
46	41, 45	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 2568  ∀wral 3052  Vcvv 3464  {csn 4606  ⟨cop 4612   class class class wbr 5124  dom cdm 5659   ↾ cres 5661  ℩cio 6487  Fun wfun 6530   Fn wfn 6531   defAt wdfat 47112  ''''cafv2 47204

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-iota 6489  df-fun 6538  df-fn 6539  df-dfat 47115  df-afv2 47205

This theorem is referenced by:  tz6.12-1-afv2  47237