Theorem funressndmafv2rn 44196

Description: The alternate function value at a class 𝐴 is defined, i.e., in the range of the function if the function is defined at 𝐴. (Contributed by AV, 2-Sep-2022.)

Assertion

Ref	Expression
funressndmafv2rn	⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)

Proof of Theorem funressndmafv2rn

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfatafv2iota 44183	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑦𝐴𝐹𝑦))
2		df-dfat 44092	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3		sneq 4535	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	reseq2d 5828	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐴}))
5	4	funeqd 6362	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (Fun (𝐹 ↾ {𝑥}) ↔ Fun (𝐹 ↾ {𝐴})))
6		eleq1 2839	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝐴 ∈ dom 𝐹))
7	5, 6	anbi12d 633	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹)))
8		breq1 5039	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
9	8	iotabidv 6324	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝐴𝐹𝑦))
10	9	eleq1d 2836	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
11	7, 10	imbi12d 348	. . . . 5 ⊢ (𝑥 = 𝐴 → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹) ↔ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)))
12		eqid 2758	. . . . . . . . 9 ⊢ (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)
13		iotaex 6320	. . . . . . . . . 10 ⊢ (℩𝑦𝑥𝐹𝑦) ∈ V
14		eqeq2 2770	. . . . . . . . . . . 12 ⊢ (𝑧 = (℩𝑦𝑥𝐹𝑦) → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)))
15		breq2 5040	. . . . . . . . . . . 12 ⊢ (𝑧 = (℩𝑦𝑥𝐹𝑦) → (𝑥𝐹𝑧 ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
16	14, 15	bibi12d 349	. . . . . . . . . . 11 ⊢ (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ 𝑥𝐹𝑧) ↔ ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦))))
17	16	imbi2d 344	. . . . . . . . . 10 ⊢ (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ 𝑥𝐹𝑧)) ↔ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))))
18		eldmg 5744	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ dom 𝐹 → (𝑥 ∈ dom 𝐹 ↔ ∃𝑧 𝑥𝐹𝑧))
19	18	ibi 270	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝐹 → ∃𝑧 𝑥𝐹𝑧)
20	19	adantl 485	. . . . . . . . . . . 12 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧 𝑥𝐹𝑧)
21		funressnvmo 44038	. . . . . . . . . . . . . 14 ⊢ (Fun (𝐹 ↾ {𝑥}) → ∃*𝑧 𝑥𝐹𝑧)
22	21	adantr 484	. . . . . . . . . . . . 13 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃*𝑧 𝑥𝐹𝑧)
23		moeu 2602	. . . . . . . . . . . . 13 ⊢ (∃*𝑧 𝑥𝐹𝑧 ↔ (∃𝑧 𝑥𝐹𝑧 → ∃!𝑧 𝑥𝐹𝑧))
24	22, 23	sylib 221	. . . . . . . . . . . 12 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (∃𝑧 𝑥𝐹𝑧 → ∃!𝑧 𝑥𝐹𝑧))
25	20, 24	mpd 15	. . . . . . . . . . 11 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃!𝑧 𝑥𝐹𝑧)
26		iota1 6317	. . . . . . . . . . . 12 ⊢ (∃!𝑧 𝑥𝐹𝑧 → (𝑥𝐹𝑧 ↔ (℩𝑧𝑥𝐹𝑧) = 𝑧))
27		breq2 5040	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → (𝑥𝐹𝑧 ↔ 𝑥𝐹𝑦))
28	27	cbviotavw 6307	. . . . . . . . . . . . 13 ⊢ (℩𝑧𝑥𝐹𝑧) = (℩𝑦𝑥𝐹𝑦)
29	28	eqeq1i 2763	. . . . . . . . . . . 12 ⊢ ((℩𝑧𝑥𝐹𝑧) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = 𝑧)
30	26, 29	bitr2di 291	. . . . . . . . . . 11 ⊢ (∃!𝑧 𝑥𝐹𝑧 → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ 𝑥𝐹𝑧))
31	25, 30	syl 17	. . . . . . . . . 10 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ 𝑥𝐹𝑧))
32	13, 17, 31	vtocl 3479	. . . . . . . . 9 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
33	12, 32	mpbii 236	. . . . . . . 8 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → 𝑥𝐹(℩𝑦𝑥𝐹𝑦))
34		df-br 5037	. . . . . . . 8 ⊢ (𝑥𝐹(℩𝑦𝑥𝐹𝑦) ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
35	33, 34	sylib 221	. . . . . . 7 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
36		vex 3413	. . . . . . . 8 ⊢ 𝑥 ∈ V
37		opeq1 4764	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ = ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩)
38	37	eleq1d 2836	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹 ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹))
39	36, 38	spcev 3527	. . . . . . 7 ⊢ (⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹 → ∃𝑧⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
40	35, 39	syl 17	. . . . . 6 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
41	13	elrn2 5738	. . . . . 6 ⊢ ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ ∃𝑧⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
42	40, 41	sylibr 237	. . . . 5 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹)
43	11, 42	vtoclg 3487	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
44	43	anabsi6 669	. . 3 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)
45	2, 44	sylbi 220	. 2 ⊢ (𝐹 defAt 𝐴 → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)
46	1, 45	eqeltrd 2852	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃*wmo 2555  ∃!weu 2587  {csn 4525  ⟨cop 4531   class class class wbr 5036  dom cdm 5528  ran crn 5529   ↾ cres 5530  ℩cio 6297  Fun wfun 6334   defAt wdfat 44089  ''''cafv2 44181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-iota 6299  df-fun 6342  df-dfat 44092  df-afv2 44182

This theorem is referenced by:  afv2ndefb  44197  dfatafv2rnb  44200