Theorem tz6.12-afv2 44732

Description: Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6797. (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 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ V)
2		vex 3436	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
3	2	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ V)
4		df-br 5075	. . . . . . . . . . 11 ⊢ (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
5	4	biimpri 227	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝑦⟩ ∈ 𝐹 → 𝐴𝐹𝑦)
6	5	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴𝐹𝑦)
7		breldmg 5818	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑦 ∈ V ∧ 𝐴𝐹𝑦) → 𝐴 ∈ dom 𝐹)
8	1, 3, 6, 7	syl3anc 1370	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
9		simpl 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → 𝐴 ∈ dom 𝐹)
10		velsn 4577	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11		breq1 5077	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 = 𝑥 → (𝐴𝐹𝑦 ↔ 𝑥𝐹𝑦))
12	4, 11	bitr3id 285	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 = 𝑥 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ 𝑥𝐹𝑦))
13	12	eqcoms 2746	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ 𝑥𝐹𝑦))
14	13	eubidv 2586	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦 𝑥𝐹𝑦))
15	14	biimpd 228	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 → ∃!𝑦 𝑥𝐹𝑦))
16	10, 15	sylbi 216	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} → ∃!𝑦 𝑥𝐹𝑦))
19	18	ralrimiv 3102	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
20		fnres 6559	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ {𝐴}) Fn {𝐴} ↔ ∀𝑥 ∈ {𝐴}∃!𝑦 𝑥𝐹𝑦)
21		fnfun 6533	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ {𝐴}) Fn {𝐴} → Fun (𝐹 ↾ {𝐴}))
22	20, 21	sylbir 234	. . . . . . . . . . 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 512	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
25	24	ex 413	. . . . . . . 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 455	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
28		df-dfat 44611	. . . . . 6 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
29	27, 28	sylibr 233	. . . . 5 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → 𝐹 defAt 𝐴)
30		dfatafv2iota 44702	. . . . 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 223	. . . . . . . . 9 ⊢ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ 𝐴𝐹𝑦)
33	32	eubii 2585	. . . . . . . 8 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦 𝐴𝐹𝑦)
34	33	biimpi 215	. . . . . . 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 482	. . . . 5 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦))
37		iota1 6410	. . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦))
38	37	biimpac 479	. . . . 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 2778	. . 3 ⊢ ((𝐴 ∈ V ∧ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)) → (𝐹''''𝐴) = 𝑦)
41	40	ex 413	. 2 ⊢ (𝐴 ∈ V → ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦))
42		eu2ndop1stv 44617	. . . . 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 482	. . 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 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃!weu 2568  ∀wral 3064  Vcvv 3432  {csn 4561  ⟨cop 4567   class class class wbr 5074  dom cdm 5589   ↾ cres 5591  ℩cio 6389  Fun wfun 6427   Fn wfn 6428   defAt wdfat 44608  ''''cafv2 44700

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fn 6436  df-dfat 44611  df-afv2 44701

This theorem is referenced by:  tz6.12-1-afv2  44733