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Theorem funressndmafv2rn 43284
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 43271 . 2 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑦𝐴𝐹𝑦))
2 df-dfat 43180 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3 sneq 4573 . . . . . . . . 9 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43reseq2d 5851 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐴}))
54funeqd 6373 . . . . . . 7 (𝑥 = 𝐴 → (Fun (𝐹 ↾ {𝑥}) ↔ Fun (𝐹 ↾ {𝐴})))
6 eleq1 2904 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹𝐴 ∈ dom 𝐹))
75, 6anbi12d 630 . . . . . 6 (𝑥 = 𝐴 → ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹)))
8 breq1 5065 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
98iotabidv 6336 . . . . . . 7 (𝑥 = 𝐴 → (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝐴𝐹𝑦))
109eleq1d 2901 . . . . . 6 (𝑥 = 𝐴 → ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
117, 10imbi12d 346 . . . . 5 (𝑥 = 𝐴 → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹) ↔ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)))
12 eqid 2825 . . . . . . . . 9 (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)
13 iotaex 6332 . . . . . . . . . 10 (℩𝑦𝑥𝐹𝑦) ∈ V
14 eqeq2 2837 . . . . . . . . . . . 12 (𝑧 = (℩𝑦𝑥𝐹𝑦) → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)))
15 breq2 5066 . . . . . . . . . . . 12 (𝑧 = (℩𝑦𝑥𝐹𝑦) → (𝑥𝐹𝑧𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
1614, 15bibi12d 347 . . . . . . . . . . 11 (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧) ↔ ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦))))
1716imbi2d 342 . . . . . . . . . 10 (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧)) ↔ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))))
18 eldmg 5765 . . . . . . . . . . . . . 14 (𝑥 ∈ dom 𝐹 → (𝑥 ∈ dom 𝐹 ↔ ∃𝑧 𝑥𝐹𝑧))
1918ibi 268 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝐹 → ∃𝑧 𝑥𝐹𝑧)
2019adantl 482 . . . . . . . . . . . 12 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧 𝑥𝐹𝑧)
21 funressnvmo 43142 . . . . . . . . . . . . . 14 (Fun (𝐹 ↾ {𝑥}) → ∃*𝑧 𝑥𝐹𝑧)
2221adantr 481 . . . . . . . . . . . . 13 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃*𝑧 𝑥𝐹𝑧)
23 moeu 2665 . . . . . . . . . . . . 13 (∃*𝑧 𝑥𝐹𝑧 ↔ (∃𝑧 𝑥𝐹𝑧 → ∃!𝑧 𝑥𝐹𝑧))
2422, 23sylib 219 . . . . . . . . . . . 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 6329 . . . . . . . . . . . 12 (∃!𝑧 𝑥𝐹𝑧 → (𝑥𝐹𝑧 ↔ (℩𝑧𝑥𝐹𝑧) = 𝑧))
27 breq2 5066 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → (𝑥𝐹𝑧𝑥𝐹𝑦))
2827cbviotav 6321 . . . . . . . . . . . . 13 (℩𝑧𝑥𝐹𝑧) = (℩𝑦𝑥𝐹𝑦)
2928eqeq1i 2830 . . . . . . . . . . . 12 ((℩𝑧𝑥𝐹𝑧) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = 𝑧)
3026, 29syl6rbb 289 . . . . . . . . . . 11 (∃!𝑧 𝑥𝐹𝑧 → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧))
3125, 30syl 17 . . . . . . . . . 10 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧))
3213, 17, 31vtocl 3564 . . . . . . . . 9 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
3312, 32mpbii 234 . . . . . . . 8 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → 𝑥𝐹(℩𝑦𝑥𝐹𝑦))
34 df-br 5063 . . . . . . . 8 (𝑥𝐹(℩𝑦𝑥𝐹𝑦) ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
3533, 34sylib 219 . . . . . . 7 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
36 vex 3502 . . . . . . . 8 𝑥 ∈ V
37 opeq1 4801 . . . . . . . . 9 (𝑧 = 𝑥 → ⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ = ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩)
3837eleq1d 2901 . . . . . . . 8 (𝑧 = 𝑥 → (⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹 ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹))
3936, 38spcev 3610 . . . . . . 7 (⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹 → ∃𝑧𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
4035, 39syl 17 . . . . . 6 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
4113elrn2 5819 . . . . . 6 ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ ∃𝑧𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
4240, 41sylibr 235 . . . . 5 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹)
4311, 42vtoclg 3572 . . . 4 (𝐴 ∈ dom 𝐹 → ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
4443anabsi6 666 . . 3 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)
452, 44sylbi 218 . 2 (𝐹 defAt 𝐴 → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)
461, 45eqeltrd 2917 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  ∃*wmo 2617  ∃!weu 2650  {csn 4563  cop 4569   class class class wbr 5062  dom cdm 5553  ran crn 5554  cres 5555  cio 6309  Fun wfun 6345   defAt wdfat 43177  ''''cafv2 43269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-iota 6311  df-fun 6353  df-dfat 43180  df-afv2 43270
This theorem is referenced by:  afv2ndefb  43285  dfatafv2rnb  43288
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