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Theorem dfatbrafv2b 42087
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6458 or funbrfvb 6460. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 42055). (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 2797 . . . 4 (𝐹''''𝐴) = (𝐹''''𝐴)
2 dfatafv2ex 42055 . . . . . 6 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)
32adantr 473 . . . . 5 ((𝐹 defAt 𝐴𝐵𝑊) → (𝐹''''𝐴) ∈ V)
4 eqeq2 2808 . . . . . . 7 (𝑥 = (𝐹''''𝐴) → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = (𝐹''''𝐴)))
5 breq2 4845 . . . . . . 7 (𝑥 = (𝐹''''𝐴) → (𝐴𝐹𝑥𝐴𝐹(𝐹''''𝐴)))
64, 5bibi12d 337 . . . . . 6 (𝑥 = (𝐹''''𝐴) → (((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥) ↔ ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴))))
76adantl 474 . . . . 5 (((𝐹 defAt 𝐴𝐵𝑊) ∧ 𝑥 = (𝐹''''𝐴)) → (((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥) ↔ ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴))))
8 dfdfat2 41970 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
9 tz6.12c-afv2 42084 . . . . . . 7 (∃!𝑥 𝐴𝐹𝑥 → ((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥))
108, 9simplbiim 500 . . . . . 6 (𝐹 defAt 𝐴 → ((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥))
1110adantr 473 . . . . 5 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥))
123, 7, 11vtocld 3442 . . . 4 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴)))
131, 12mpbii 225 . . 3 ((𝐹 defAt 𝐴𝐵𝑊) → 𝐴𝐹(𝐹''''𝐴))
14 breq2 4845 . . 3 ((𝐹''''𝐴) = 𝐵 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝐵))
1513, 14syl5ibcom 237 . 2 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))
16 df-dfat 41961 . . . 4 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
17 simpll 784 . . . . 5 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵𝑊) → 𝐴 ∈ dom 𝐹)
18 simpr 478 . . . . 5 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵𝑊) → 𝐵𝑊)
19 simpr 478 . . . . . 6 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → Fun (𝐹 ↾ {𝐴}))
2019adantr 473 . . . . 5 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵𝑊) → Fun (𝐹 ↾ {𝐴}))
2117, 18, 20jca31 511 . . . 4 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵𝑊) → ((𝐴 ∈ dom 𝐹𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})))
2216, 21sylanb 577 . . 3 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐴 ∈ dom 𝐹𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})))
23 funressnbrafv2 42086 . . 3 (((𝐴 ∈ dom 𝐹𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
2422, 23syl 17 . 2 ((𝐹 defAt 𝐴𝐵𝑊) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
2515, 24impbid 204 1 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  ∃!weu 2606  Vcvv 3383  {csn 4366   class class class wbr 4841  dom cdm 5310  cres 5312  Fun wfun 6093   defAt wdfat 41958  ''''cafv2 42050
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-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095
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 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-res 5322  df-iota 6062  df-fun 6101  df-fn 6102  df-dfat 41961  df-afv2 42051
This theorem is referenced by:  dfatopafv2b  42088  dfatsnafv2  42094  dfatcolem  42097
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