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Theorem funressndmafv2rn 46231
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 46218 . 2 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑦𝐴𝐹𝑦))
2 df-dfat 46127 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3 sneq 4639 . . . . . . . . 9 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43reseq2d 5982 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐴}))
54funeqd 6571 . . . . . . 7 (𝑥 = 𝐴 → (Fun (𝐹 ↾ {𝑥}) ↔ Fun (𝐹 ↾ {𝐴})))
6 eleq1 2819 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹𝐴 ∈ dom 𝐹))
75, 6anbi12d 629 . . . . . 6 (𝑥 = 𝐴 → ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹)))
8 breq1 5152 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
98iotabidv 6528 . . . . . . 7 (𝑥 = 𝐴 → (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝐴𝐹𝑦))
109eleq1d 2816 . . . . . 6 (𝑥 = 𝐴 → ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
117, 10imbi12d 343 . . . . 5 (𝑥 = 𝐴 → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹) ↔ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)))
12 eqid 2730 . . . . . . . . 9 (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)
13 iotaex 6517 . . . . . . . . . 10 (℩𝑦𝑥𝐹𝑦) ∈ V
14 eqeq2 2742 . . . . . . . . . . . 12 (𝑧 = (℩𝑦𝑥𝐹𝑦) → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)))
15 breq2 5153 . . . . . . . . . . . 12 (𝑧 = (℩𝑦𝑥𝐹𝑦) → (𝑥𝐹𝑧𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
1614, 15bibi12d 344 . . . . . . . . . . 11 (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧) ↔ ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦))))
1716imbi2d 339 . . . . . . . . . 10 (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧)) ↔ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))))
18 eldmg 5899 . . . . . . . . . . . . . 14 (𝑥 ∈ dom 𝐹 → (𝑥 ∈ dom 𝐹 ↔ ∃𝑧 𝑥𝐹𝑧))
1918ibi 266 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝐹 → ∃𝑧 𝑥𝐹𝑧)
2019adantl 480 . . . . . . . . . . . 12 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧 𝑥𝐹𝑧)
21 funressnvmo 46055 . . . . . . . . . . . . . 14 (Fun (𝐹 ↾ {𝑥}) → ∃*𝑧 𝑥𝐹𝑧)
2221adantr 479 . . . . . . . . . . . . 13 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃*𝑧 𝑥𝐹𝑧)
23 moeu 2575 . . . . . . . . . . . . 13 (∃*𝑧 𝑥𝐹𝑧 ↔ (∃𝑧 𝑥𝐹𝑧 → ∃!𝑧 𝑥𝐹𝑧))
2422, 23sylib 217 . . . . . . . . . . . 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 6521 . . . . . . . . . . . 12 (∃!𝑧 𝑥𝐹𝑧 → (𝑥𝐹𝑧 ↔ (℩𝑧𝑥𝐹𝑧) = 𝑧))
27 breq2 5153 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → (𝑥𝐹𝑧𝑥𝐹𝑦))
2827cbviotavw 6504 . . . . . . . . . . . . 13 (℩𝑧𝑥𝐹𝑧) = (℩𝑦𝑥𝐹𝑦)
2928eqeq1i 2735 . . . . . . . . . . . 12 ((℩𝑧𝑥𝐹𝑧) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = 𝑧)
3026, 29bitr2di 287 . . . . . . . . . . 11 (∃!𝑧 𝑥𝐹𝑧 → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧))
3125, 30syl 17 . . . . . . . . . 10 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧))
3213, 17, 31vtocl 3544 . . . . . . . . 9 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
3312, 32mpbii 232 . . . . . . . 8 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → 𝑥𝐹(℩𝑦𝑥𝐹𝑦))
34 df-br 5150 . . . . . . . 8 (𝑥𝐹(℩𝑦𝑥𝐹𝑦) ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
3533, 34sylib 217 . . . . . . 7 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
36 vex 3476 . . . . . . . 8 𝑥 ∈ V
37 opeq1 4874 . . . . . . . . 9 (𝑧 = 𝑥 → ⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ = ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩)
3837eleq1d 2816 . . . . . . . 8 (𝑧 = 𝑥 → (⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹 ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹))
3936, 38spcev 3597 . . . . . . 7 (⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹 → ∃𝑧𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
4035, 39syl 17 . . . . . 6 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
4113elrn2 5893 . . . . . 6 ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ ∃𝑧𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
4240, 41sylibr 233 . . . . 5 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹)
4311, 42vtoclg 3541 . . . 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 216 . 2 (𝐹 defAt 𝐴 → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)
461, 45eqeltrd 2831 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wex 1779  wcel 2104  ∃*wmo 2530  ∃!weu 2560  {csn 4629  cop 4635   class class class wbr 5149  dom cdm 5677  ran crn 5678  cres 5679  cio 6494  Fun wfun 6538   defAt wdfat 46124  ''''cafv2 46216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-dfat 46127  df-afv2 46217
This theorem is referenced by:  afv2ndefb  46232  dfatafv2rnb  46235
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