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Theorem funressndmafv2rn 45931
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 45918 . 2 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑦𝐴𝐹𝑦))
2 df-dfat 45827 . . 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 2822 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹𝐴 ∈ dom 𝐹))
75, 6anbi12d 632 . . . . . 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 2819 . . . . . 6 (𝑥 = 𝐴 → ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
117, 10imbi12d 345 . . . . 5 (𝑥 = 𝐴 → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹) ↔ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)))
12 eqid 2733 . . . . . . . . 9 (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)
13 iotaex 6517 . . . . . . . . . 10 (℩𝑦𝑥𝐹𝑦) ∈ V
14 eqeq2 2745 . . . . . . . . . . . 12 (𝑧 = (℩𝑦𝑥𝐹𝑦) → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)))
15 breq2 5153 . . . . . . . . . . . 12 (𝑧 = (℩𝑦𝑥𝐹𝑦) → (𝑥𝐹𝑧𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
1614, 15bibi12d 346 . . . . . . . . . . 11 (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧) ↔ ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦))))
1716imbi2d 341 . . . . . . . . . 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 267 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝐹 → ∃𝑧 𝑥𝐹𝑧)
2019adantl 483 . . . . . . . . . . . 12 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧 𝑥𝐹𝑧)
21 funressnvmo 45755 . . . . . . . . . . . . . 14 (Fun (𝐹 ↾ {𝑥}) → ∃*𝑧 𝑥𝐹𝑧)
2221adantr 482 . . . . . . . . . . . . 13 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃*𝑧 𝑥𝐹𝑧)
23 moeu 2578 . . . . . . . . . . . . 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 2738 . . . . . . . . . . . 12 ((℩𝑧𝑥𝐹𝑧) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = 𝑧)
3026, 29bitr2di 288 . . . . . . . . . . 11 (∃!𝑧 𝑥𝐹𝑧 → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧))
3125, 30syl 17 . . . . . . . . . 10 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧))
3213, 17, 31vtocl 3550 . . . . . . . . 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 3479 . . . . . . . 8 𝑥 ∈ V
37 opeq1 4874 . . . . . . . . 9 (𝑧 = 𝑥 → ⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ = ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩)
3837eleq1d 2819 . . . . . . . 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 3557 . . . 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 216 . 2 (𝐹 defAt 𝐴 → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)
461, 45eqeltrd 2834 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  ∃*wmo 2533  ∃!weu 2563  {csn 4629  cop 4635   class class class wbr 5149  dom cdm 5677  ran crn 5678  cres 5679  cio 6494  Fun wfun 6538   defAt wdfat 45824  ''''cafv2 45916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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 45827  df-afv2 45917
This theorem is referenced by:  afv2ndefb  45932  dfatafv2rnb  45935
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