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Theorem dfatbrafv2b 43307
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6714 or funbrfvb 6716. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 43275). (Contributed by AV, 6-Sep-2022.)
Assertion
Ref Expression
dfatbrafv2b ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))

Proof of Theorem dfatbrafv2b
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . . . 4 (𝐹''''𝐴) = (𝐹''''𝐴)
2 dfatafv2ex 43275 . . . . . 6 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)
32adantr 481 . . . . 5 ((𝐹 defAt 𝐴𝐵𝑊) → (𝐹''''𝐴) ∈ V)
4 eqeq2 2836 . . . . . . 7 (𝑥 = (𝐹''''𝐴) → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = (𝐹''''𝐴)))
5 breq2 5066 . . . . . . 7 (𝑥 = (𝐹''''𝐴) → (𝐴𝐹𝑥𝐴𝐹(𝐹''''𝐴)))
64, 5bibi12d 347 . . . . . 6 (𝑥 = (𝐹''''𝐴) → (((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥) ↔ ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴))))
76adantl 482 . . . . 5 (((𝐹 defAt 𝐴𝐵𝑊) ∧ 𝑥 = (𝐹''''𝐴)) → (((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥) ↔ ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴))))
8 dfdfat2 43190 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
9 tz6.12c-afv2 43304 . . . . . . 7 (∃!𝑥 𝐴𝐹𝑥 → ((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥))
108, 9simplbiim 505 . . . . . 6 (𝐹 defAt 𝐴 → ((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥))
1110adantr 481 . . . . 5 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥))
123, 7, 11vtocld 3561 . . . 4 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴)))
131, 12mpbii 234 . . 3 ((𝐹 defAt 𝐴𝐵𝑊) → 𝐴𝐹(𝐹''''𝐴))
14 breq2 5066 . . 3 ((𝐹''''𝐴) = 𝐵 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝐵))
1513, 14syl5ibcom 246 . 2 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))
16 df-dfat 43181 . . . 4 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
17 simpll 763 . . . . 5 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵𝑊) → 𝐴 ∈ dom 𝐹)
18 simpr 485 . . . . 5 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵𝑊) → 𝐵𝑊)
19 simpr 485 . . . . . 6 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → Fun (𝐹 ↾ {𝐴}))
2019adantr 481 . . . . 5 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵𝑊) → Fun (𝐹 ↾ {𝐴}))
2117, 18, 20jca31 515 . . . 4 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵𝑊) → ((𝐴 ∈ dom 𝐹𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})))
2216, 21sylanb 581 . . 3 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐴 ∈ dom 𝐹𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})))
23 funressnbrafv2 43306 . . 3 (((𝐴 ∈ dom 𝐹𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
2422, 23syl 17 . 2 ((𝐹 defAt 𝐴𝐵𝑊) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
2515, 24impbid 213 1 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2106  ∃!weu 2648  Vcvv 3499  {csn 4563   class class class wbr 5062  dom cdm 5553  cres 5555  Fun wfun 6345   defAt wdfat 43178  ''''cafv2 43270
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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  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 2615  df-eu 2649  df-clab 2803  df-cleq 2817  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-res 5565  df-iota 6311  df-fun 6353  df-fn 6354  df-dfat 43181  df-afv2 43271
This theorem is referenced by:  dfatopafv2b  43308  dfatsnafv2  43314  dfatcolem  43317
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