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Theorem dfatbrafv2b 44737
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6822 or funbrfvb 6824. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 44705). (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 2738 . . . 4 (𝐹''''𝐴) = (𝐹''''𝐴)
2 dfatafv2ex 44705 . . . . . 6 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)
32adantr 481 . . . . 5 ((𝐹 defAt 𝐴𝐵𝑊) → (𝐹''''𝐴) ∈ V)
4 eqeq2 2750 . . . . . . 7 (𝑥 = (𝐹''''𝐴) → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = (𝐹''''𝐴)))
5 breq2 5078 . . . . . . 7 (𝑥 = (𝐹''''𝐴) → (𝐴𝐹𝑥𝐴𝐹(𝐹''''𝐴)))
64, 5bibi12d 346 . . . . . 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 44620 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
9 tz6.12c-afv2 44734 . . . . . . 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 3494 . . . 4 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴)))
131, 12mpbii 232 . . 3 ((𝐹 defAt 𝐴𝐵𝑊) → 𝐴𝐹(𝐹''''𝐴))
14 breq2 5078 . . 3 ((𝐹''''𝐴) = 𝐵 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝐵))
1513, 14syl5ibcom 244 . 2 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))
16 df-dfat 44611 . . . 4 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
17 simpll 764 . . . . 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 44736 . . 3 (((𝐴 ∈ dom 𝐹𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
2422, 23syl 17 . 2 ((𝐹 defAt 𝐴𝐵𝑊) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
2515, 24impbid 211 1 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  ∃!weu 2568  Vcvv 3432  {csn 4561   class class class wbr 5074  dom cdm 5589  cres 5591  Fun wfun 6427   defAt wdfat 44608  ''''cafv2 44700
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fn 6436  df-dfat 44611  df-afv2 44701
This theorem is referenced by:  dfatopafv2b  44738  dfatsnafv2  44744  dfatcolem  44747
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