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Theorem dfatbrafv2b 44317
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6734 or funbrfvb 6736. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 44285). (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 2739 . . . 4 (𝐹''''𝐴) = (𝐹''''𝐴)
2 dfatafv2ex 44285 . . . . . 6 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)
32adantr 484 . . . . 5 ((𝐹 defAt 𝐴𝐵𝑊) → (𝐹''''𝐴) ∈ V)
4 eqeq2 2751 . . . . . . 7 (𝑥 = (𝐹''''𝐴) → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = (𝐹''''𝐴)))
5 breq2 5044 . . . . . . 7 (𝑥 = (𝐹''''𝐴) → (𝐴𝐹𝑥𝐴𝐹(𝐹''''𝐴)))
64, 5bibi12d 349 . . . . . 6 (𝑥 = (𝐹''''𝐴) → (((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥) ↔ ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴))))
76adantl 485 . . . . 5 (((𝐹 defAt 𝐴𝐵𝑊) ∧ 𝑥 = (𝐹''''𝐴)) → (((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥) ↔ ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴))))
8 dfdfat2 44200 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
9 tz6.12c-afv2 44314 . . . . . . 7 (∃!𝑥 𝐴𝐹𝑥 → ((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥))
108, 9simplbiim 508 . . . . . 6 (𝐹 defAt 𝐴 → ((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥))
1110adantr 484 . . . . 5 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = 𝑥𝐴𝐹𝑥))
123, 7, 11vtocld 3462 . . . 4 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴)))
131, 12mpbii 236 . . 3 ((𝐹 defAt 𝐴𝐵𝑊) → 𝐴𝐹(𝐹''''𝐴))
14 breq2 5044 . . 3 ((𝐹''''𝐴) = 𝐵 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝐵))
1513, 14syl5ibcom 248 . 2 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))
16 df-dfat 44191 . . . 4 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
17 simpll 767 . . . . 5 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵𝑊) → 𝐴 ∈ dom 𝐹)
18 simpr 488 . . . . 5 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵𝑊) → 𝐵𝑊)
19 simpr 488 . . . . . 6 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → Fun (𝐹 ↾ {𝐴}))
2019adantr 484 . . . . 5 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵𝑊) → Fun (𝐹 ↾ {𝐴}))
2117, 18, 20jca31 518 . . . 4 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵𝑊) → ((𝐴 ∈ dom 𝐹𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})))
2216, 21sylanb 584 . . 3 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐴 ∈ dom 𝐹𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})))
23 funressnbrafv2 44316 . . 3 (((𝐴 ∈ dom 𝐹𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
2422, 23syl 17 . 2 ((𝐹 defAt 𝐴𝐵𝑊) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
2515, 24impbid 215 1 ((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  ∃!weu 2570  Vcvv 3400  {csn 4526   class class class wbr 5040  dom cdm 5535  cres 5537  Fun wfun 6343   defAt wdfat 44188  ''''cafv2 44280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-res 5547  df-iota 6307  df-fun 6351  df-fn 6352  df-dfat 44191  df-afv2 44281
This theorem is referenced by:  dfatopafv2b  44318  dfatsnafv2  44324  dfatcolem  44327
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