Theorem funressndmafv2rn 47411

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 47398	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑦𝐴𝐹𝑦))
2		df-dfat 47307	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3		sneq 4588	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	reseq2d 5936	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐴}))
5	4	funeqd 6512	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (Fun (𝐹 ↾ {𝑥}) ↔ Fun (𝐹 ↾ {𝐴})))
6		eleq1 2822	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝐴 ∈ dom 𝐹))
7	5, 6	anbi12d 632	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹)))
8		breq1 5099	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
9	8	iotabidv 6474	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝐴𝐹𝑦))
10	9	eleq1d 2819	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
11	7, 10	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝐴 → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹) ↔ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)))
12		eqid 2734	. . . . . . . . 9 ⊢ (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)
13		iotaex 6466	. . . . . . . . . 10 ⊢ (℩𝑦𝑥𝐹𝑦) ∈ V
14		eqeq2 2746	. . . . . . . . . . . 12 ⊢ (𝑧 = (℩𝑦𝑥𝐹𝑦) → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)))
15		breq2 5100	. . . . . . . . . . . 12 ⊢ (𝑧 = (℩𝑦𝑥𝐹𝑦) → (𝑥𝐹𝑧 ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
16	14, 15	bibi12d 345	. . . . . . . . . . 11 ⊢ (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ 𝑥𝐹𝑧) ↔ ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦))))
17	16	imbi2d 340	. . . . . . . . . 10 ⊢ (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ 𝑥𝐹𝑧)) ↔ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))))
18		eldmg 5845	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ dom 𝐹 → (𝑥 ∈ dom 𝐹 ↔ ∃𝑧 𝑥𝐹𝑧))
19	18	ibi 267	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝐹 → ∃𝑧 𝑥𝐹𝑧)
20	19	adantl 481	. . . . . . . . . . . 12 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧 𝑥𝐹𝑧)
21		funressnvmo 47233	. . . . . . . . . . . . . 14 ⊢ (Fun (𝐹 ↾ {𝑥}) → ∃*𝑧 𝑥𝐹𝑧)
22	21	adantr 480	. . . . . . . . . . . . 13 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃*𝑧 𝑥𝐹𝑧)
23		moeu 2581	. . . . . . . . . . . . 13 ⊢ (∃*𝑧 𝑥𝐹𝑧 ↔ (∃𝑧 𝑥𝐹𝑧 → ∃!𝑧 𝑥𝐹𝑧))
24	22, 23	sylib 218	. . . . . . . . . . . 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 6469	. . . . . . . . . . . 12 ⊢ (∃!𝑧 𝑥𝐹𝑧 → (𝑥𝐹𝑧 ↔ (℩𝑧𝑥𝐹𝑧) = 𝑧))
27		breq2 5100	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → (𝑥𝐹𝑧 ↔ 𝑥𝐹𝑦))
28	27	cbviotavw 6454	. . . . . . . . . . . . 13 ⊢ (℩𝑧𝑥𝐹𝑧) = (℩𝑦𝑥𝐹𝑦)
29	28	eqeq1i 2739	. . . . . . . . . . . 12 ⊢ ((℩𝑧𝑥𝐹𝑧) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = 𝑧)
30	26, 29	bitr2di 288	. . . . . . . . . . 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 3513	. . . . . . . . 9 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
33	12, 32	mpbii 233	. . . . . . . 8 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → 𝑥𝐹(℩𝑦𝑥𝐹𝑦))
34		df-br 5097	. . . . . . . 8 ⊢ (𝑥𝐹(℩𝑦𝑥𝐹𝑦) ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
35	33, 34	sylib 218	. . . . . . 7 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
36		vex 3442	. . . . . . . 8 ⊢ 𝑥 ∈ V
37		opeq1 4827	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ = ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩)
38	37	eleq1d 2819	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹 ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹))
39	36, 38	spcev 3558	. . . . . . 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 5839	. . . . . 6 ⊢ ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ ∃𝑧⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
42	40, 41	sylibr 234	. . . . 5 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹)
43	11, 42	vtoclg 3509	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
44	43	anabsi6 670	. . 3 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)
45	2, 44	sylbi 217	. 2 ⊢ (𝐹 defAt 𝐴 → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)
46	1, 45	eqeltrd 2834	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃*wmo 2535  ∃!weu 2566  {csn 4578  ⟨cop 4584   class class class wbr 5096  dom cdm 5622  ran crn 5623   ↾ cres 5624  ℩cio 6444  Fun wfun 6484   defAt wdfat 47304  ''''cafv2 47396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-iota 6446  df-fun 6492  df-dfat 47307  df-afv2 47397

This theorem is referenced by:  afv2ndefb  47412  dfatafv2rnb  47415