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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.
StepHypRef 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})
43reseq2d 5828 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐴}))
54funeqd 6362 . . . . . . 7 (𝑥 = 𝐴 → (Fun (𝐹 ↾ {𝑥}) ↔ Fun (𝐹 ↾ {𝐴})))
6 eleq1 2839 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹𝐴 ∈ dom 𝐹))
75, 6anbi12d 633 . . . . . 6 (𝑥 = 𝐴 → ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹)))
8 breq1 5039 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
98iotabidv 6324 . . . . . . 7 (𝑥 = 𝐴 → (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝐴𝐹𝑦))
109eleq1d 2836 . . . . . 6 (𝑥 = 𝐴 → ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
117, 10imbi12d 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) → (𝑥𝐹𝑧𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
1614, 15bibi12d 349 . . . . . . . . . . 11 (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧) ↔ ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦))))
1716imbi2d 344 . . . . . . . . . 10 (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧)) ↔ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))))
18 eldmg 5744 . . . . . . . . . . . . . 14 (𝑥 ∈ dom 𝐹 → (𝑥 ∈ dom 𝐹 ↔ ∃𝑧 𝑥𝐹𝑧))
1918ibi 270 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝐹 → ∃𝑧 𝑥𝐹𝑧)
2019adantl 485 . . . . . . . . . . . 12 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧 𝑥𝐹𝑧)
21 funressnvmo 44038 . . . . . . . . . . . . . 14 (Fun (𝐹 ↾ {𝑥}) → ∃*𝑧 𝑥𝐹𝑧)
2221adantr 484 . . . . . . . . . . . . 13 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃*𝑧 𝑥𝐹𝑧)
23 moeu 2602 . . . . . . . . . . . . 13 (∃*𝑧 𝑥𝐹𝑧 ↔ (∃𝑧 𝑥𝐹𝑧 → ∃!𝑧 𝑥𝐹𝑧))
2422, 23sylib 221 . . . . . . . . . . . 12 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (∃𝑧 𝑥𝐹𝑧 → ∃!𝑧 𝑥𝐹𝑧))
2520, 24mpd 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 → (𝑥𝐹𝑧𝑥𝐹𝑦))
2827cbviotavw 6307 . . . . . . . . . . . . 13 (℩𝑧𝑥𝐹𝑧) = (℩𝑦𝑥𝐹𝑦)
2928eqeq1i 2763 . . . . . . . . . . . 12 ((℩𝑧𝑥𝐹𝑧) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = 𝑧)
3026, 29bitr2di 291 . . . . . . . . . . 11 (∃!𝑧 𝑥𝐹𝑧 → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧))
3125, 30syl 17 . . . . . . . . . 10 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧))
3213, 17, 31vtocl 3479 . . . . . . . . 9 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
3312, 32mpbii 236 . . . . . . . 8 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → 𝑥𝐹(℩𝑦𝑥𝐹𝑦))
34 df-br 5037 . . . . . . . 8 (𝑥𝐹(℩𝑦𝑥𝐹𝑦) ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
3533, 34sylib 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)⟩)
3837eleq1d 2836 . . . . . . . 8 (𝑧 = 𝑥 → (⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹 ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹))
3936, 38spcev 3527 . . . . . . 7 (⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹 → ∃𝑧𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
4035, 39syl 17 . . . . . 6 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
4113elrn2 5738 . . . . . 6 ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ ∃𝑧𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
4240, 41sylibr 237 . . . . 5 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹)
4311, 42vtoclg 3487 . . . 4 (𝐴 ∈ dom 𝐹 → ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
4443anabsi6 669 . . 3 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)
452, 44sylbi 220 . 2 (𝐹 defAt 𝐴 → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)
461, 45eqeltrd 2852 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
 Colors of variables: wff setvar class 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
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