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Theorem funressndmafv2rn 42066
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 42053 . 2 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑦𝐴𝐹𝑦))
2 df-dfat 41962 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3 sneq 4377 . . . . . . . . 9 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43reseq2d 5599 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐴}))
54funeqd 6122 . . . . . . 7 (𝑥 = 𝐴 → (Fun (𝐹 ↾ {𝑥}) ↔ Fun (𝐹 ↾ {𝐴})))
6 eleq1 2865 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹𝐴 ∈ dom 𝐹))
75, 6anbi12d 625 . . . . . 6 (𝑥 = 𝐴 → ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹)))
8 breq1 4845 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
98iotabidv 6084 . . . . . . 7 (𝑥 = 𝐴 → (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝐴𝐹𝑦))
109eleq1d 2862 . . . . . 6 (𝑥 = 𝐴 → ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
117, 10imbi12d 336 . . . . 5 (𝑥 = 𝐴 → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹) ↔ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)))
12 eqid 2798 . . . . . . . . 9 (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)
13 iotaex 6080 . . . . . . . . . 10 (℩𝑦𝑥𝐹𝑦) ∈ V
14 eqeq2 2809 . . . . . . . . . . . 12 (𝑧 = (℩𝑦𝑥𝐹𝑦) → ((℩𝑦𝑥𝐹𝑦) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦)))
15 breq2 4846 . . . . . . . . . . . 12 (𝑧 = (℩𝑦𝑥𝐹𝑦) → (𝑥𝐹𝑧𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
1614, 15bibi12d 337 . . . . . . . . . . 11 (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧) ↔ ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦))))
1716imbi2d 332 . . . . . . . . . 10 (𝑧 = (℩𝑦𝑥𝐹𝑦) → (((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧)) ↔ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))))
18 eldmg 5521 . . . . . . . . . . . . . 14 (𝑥 ∈ dom 𝐹 → (𝑥 ∈ dom 𝐹 ↔ ∃𝑧 𝑥𝐹𝑧))
1918ibi 259 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝐹 → ∃𝑧 𝑥𝐹𝑧)
2019adantl 474 . . . . . . . . . . . 12 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧 𝑥𝐹𝑧)
21 funressnvmo 41917 . . . . . . . . . . . . . 14 (Fun (𝐹 ↾ {𝑥}) → ∃*𝑧 𝑥𝐹𝑧)
2221adantr 473 . . . . . . . . . . . . 13 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃*𝑧 𝑥𝐹𝑧)
23 moeu 2623 . . . . . . . . . . . . 13 (∃*𝑧 𝑥𝐹𝑧 ↔ (∃𝑧 𝑥𝐹𝑧 → ∃!𝑧 𝑥𝐹𝑧))
2422, 23sylib 210 . . . . . . . . . . . 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 6077 . . . . . . . . . . . 12 (∃!𝑧 𝑥𝐹𝑧 → (𝑥𝐹𝑧 ↔ (℩𝑧𝑥𝐹𝑧) = 𝑧))
27 breq2 4846 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → (𝑥𝐹𝑧𝑥𝐹𝑦))
2827cbviotav 6069 . . . . . . . . . . . . 13 (℩𝑧𝑥𝐹𝑧) = (℩𝑦𝑥𝐹𝑦)
2928eqeq1i 2803 . . . . . . . . . . . 12 ((℩𝑧𝑥𝐹𝑧) = 𝑧 ↔ (℩𝑦𝑥𝐹𝑦) = 𝑧)
3026, 29syl6rbb 280 . . . . . . . . . . 11 (∃!𝑧 𝑥𝐹𝑧 → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧))
3125, 30syl 17 . . . . . . . . . 10 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = 𝑧𝑥𝐹𝑧))
3213, 17, 31vtocl 3445 . . . . . . . . 9 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ((℩𝑦𝑥𝐹𝑦) = (℩𝑦𝑥𝐹𝑦) ↔ 𝑥𝐹(℩𝑦𝑥𝐹𝑦)))
3312, 32mpbii 225 . . . . . . . 8 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → 𝑥𝐹(℩𝑦𝑥𝐹𝑦))
34 df-br 4843 . . . . . . . 8 (𝑥𝐹(℩𝑦𝑥𝐹𝑦) ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
3533, 34sylib 210 . . . . . . 7 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
36 vex 3387 . . . . . . . 8 𝑥 ∈ V
37 opeq1 4592 . . . . . . . . 9 (𝑧 = 𝑥 → ⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ = ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩)
3837eleq1d 2862 . . . . . . . 8 (𝑧 = 𝑥 → (⟨𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹 ↔ ⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹))
3936, 38spcev 3487 . . . . . . 7 (⟨𝑥, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹 → ∃𝑧𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
4035, 39syl 17 . . . . . 6 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → ∃𝑧𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
4113elrn2 5568 . . . . . 6 ((℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹 ↔ ∃𝑧𝑧, (℩𝑦𝑥𝐹𝑦)⟩ ∈ 𝐹)
4240, 41sylibr 226 . . . . 5 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom 𝐹) → (℩𝑦𝑥𝐹𝑦) ∈ ran 𝐹)
4311, 42vtoclg 3452 . . . 4 (𝐴 ∈ dom 𝐹 → ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹))
4443anabsi6 661 . . 3 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)
452, 44sylbi 209 . 2 (𝐹 defAt 𝐴 → (℩𝑦𝐴𝐹𝑦) ∈ ran 𝐹)
461, 45eqeltrd 2877 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  ∃*wmo 2589  ∃!weu 2608  {csn 4367  cop 4373   class class class wbr 4842  dom cdm 5311  ran crn 5312  cres 5313  cio 6061  Fun wfun 6094   defAt wdfat 41959  ''''cafv2 42051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pr 5096
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3386  df-sbc 3633  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-br 4843  df-opab 4905  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-iota 6063  df-fun 6102  df-dfat 41962  df-afv2 42052
This theorem is referenced by:  afv2ndefb  42067  dfatafv2rnb  42070
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