Theorem funressndmafv2rn 48292

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 48279	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑦𝐴𝐹𝑦))
2		df-dfat 48188	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3		sneq 4594	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	reseq2d 5970	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐴}))
5	4	funeqd 6561	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (Fun (𝐹 ↾ {𝑥}) ↔ Fun (𝐹 ↾ {𝐴})))
6		eleq1 2849	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝐴 ∈ dom 𝐹))
7	5, 6	anbi12d 644	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹)))
8		breq1 5106	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
9	8	iotabidv 6522	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝐴𝐹𝑦))
10	9	eleq1d 2846	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
11	7, 10	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝐴 → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹) ↔ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)))
12		eqid 2761	. . . . . . . . 9 ⊢ (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)
13		iotaex 6514	. . . . . . . . . 10 ⊢ (℩𝑦𝑥𝐹𝑦) ∈ V
14		eqeq2 2773	. . . . . . . . . . . 12 ⊢ (𝑧 = (℩𝑦𝑥𝐹𝑦) → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)))
15		breq2 5107	. . . . . . . . . . . 12 ⊢ (𝑧 = (℩𝑦𝑥𝐹𝑦) → (𝑥𝐹𝑧 ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
16	14, 15	bibi12d 348	. . . . . . . . . . 11 ⊢ (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ 𝑥𝐹𝑧) ↔ ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦))))
17	16	imbi2d 343	. . . . . . . . . 10 ⊢ (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ 𝑥𝐹𝑧)) ↔ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))))
18		eldmg 5880	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ dom 𝐹 → (𝑥 ∈ dom 𝐹 ↔ ∃𝑧 𝑥𝐹𝑧))
19	18	ibi 270	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝐹 → ∃𝑧 𝑥𝐹𝑧)
20	19	adantl 487	. . . . . . . . . . . 12 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧 𝑥𝐹𝑧)
21		funressnvmo 48114	. . . . . . . . . . . . . 14 ⊢ (Fun (𝐹 ↾ {𝑥}) → ∃*𝑧 𝑥𝐹𝑧)
22	21	adantr 486	. . . . . . . . . . . . 13 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃*𝑧 𝑥𝐹𝑧)
23		moeu 2609	. . . . . . . . . . . . 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 16	. . . . . . . . . . 11 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃!𝑧 𝑥𝐹𝑧)
26		iota1 6517	. . . . . . . . . . . 12 ⊢ (∃!𝑧 𝑥𝐹𝑧 → (𝑥𝐹𝑧 ↔ (℩𝑧𝑥𝐹𝑧) = 𝑧))
27		breq2 5107	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → (𝑥𝐹𝑧 ↔ 𝑥𝐹𝑦))
28	27	cbviotavw 6502	. . . . . . . . . . . . 13 ⊢ (℩𝑧𝑥𝐹𝑧) = (℩𝑦𝑥𝐹𝑦)
29	28	eqeq1i 2766	. . . . . . . . . . . 12 ⊢ ((℩𝑧𝑥𝐹𝑧) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = 𝑧)
30	26, 29	bitr2di 291	. . . . . . . . . . 11 ⊢ (∃!𝑧 𝑥𝐹𝑧 → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ 𝑥𝐹𝑧))
31	25, 30	syl 18	. . . . . . . . . 10 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ 𝑥𝐹𝑧))
32	13, 17, 31	vtocl 3521	. . . . . . . . 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 5104	. . . . . . . 8 ⊢ (𝑥𝐹(℩𝑦𝑥𝐹𝑦) ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
35	33, 34	sylib 221	. . . . . . 7 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
36		vex 3455	. . . . . . . 8 ⊢ 𝑥 ∈ V
37		opeq1 4833	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ = ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩)
38	37	eleq1d 2846	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹 ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹))
39	36, 38	spcev 3561	. . . . . . 7 ⊢ (⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹 → ∃𝑧⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
40	35, 39	syl 18	. . . . . 6 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
41	13	elrn2 5874	. . . . . 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 3518	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
44	43	anabsi6 683	. . 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 2861	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2563  ∃!weu 2594  {csn 4584  ⟨cop 4590   class class class wbr 5103  dom cdm 5651  ran crn 5652   ↾ cres 5653  ℩cio 6492  Fun wfun 6532   defAt wdfat 48185  ''''cafv2 48277

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 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6494  df-fun 6540  df-dfat 48188  df-afv2 48278

This theorem is used by:  afv2ndefb  48293  dfatafv2rnb  48296