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Theorem fnbrafv2b 47260
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6959. (Contributed by AV, 6-Sep-2022.)
Assertion
Ref Expression
fnbrafv2b ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝐶𝐵𝐹𝐶))

Proof of Theorem fnbrafv2b
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . 4 (𝐹''''𝐵) = (𝐹''''𝐵)
2 fundmdfat 47141 . . . . . . 7 ((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝐹 defAt 𝐵)
32funfni 6674 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐹 defAt 𝐵)
4 dfatafv2ex 47225 . . . . . 6 (𝐹 defAt 𝐵 → (𝐹''''𝐵) ∈ V)
53, 4syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹''''𝐵) ∈ V)
6 eqeq2 2749 . . . . . . 7 (𝑥 = (𝐹''''𝐵) → ((𝐹''''𝐵) = 𝑥 ↔ (𝐹''''𝐵) = (𝐹''''𝐵)))
7 breq2 5147 . . . . . . 7 (𝑥 = (𝐹''''𝐵) → (𝐵𝐹𝑥𝐵𝐹(𝐹''''𝐵)))
86, 7bibi12d 345 . . . . . 6 (𝑥 = (𝐹''''𝐵) → (((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
98adantl 481 . . . . 5 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑥 = (𝐹''''𝐵)) → (((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
10 fneu 6678 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑥 𝐵𝐹𝑥)
11 tz6.12c-afv2 47254 . . . . . 6 (∃!𝑥 𝐵𝐹𝑥 → ((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥))
1210, 11syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥))
135, 9, 12vtocld 3561 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵)))
141, 13mpbii 233 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵𝐹(𝐹''''𝐵))
15 breq2 5147 . . 3 ((𝐹''''𝐵) = 𝐶 → (𝐵𝐹(𝐹''''𝐵) ↔ 𝐵𝐹𝐶))
1614, 15syl5ibcom 245 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝐶𝐵𝐹𝐶))
17 fnfun 6668 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
18 funbrafv2 47259 . . . 4 (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
1917, 18syl 17 . . 3 (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
2019adantr 480 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
2116, 20impbid 212 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝐶𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  ∃!weu 2568  Vcvv 3480   class class class wbr 5143  Fun wfun 6555   Fn wfn 6556   defAt wdfat 47128  ''''cafv2 47220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-dfat 47131  df-afv2 47221
This theorem is referenced by:  fnopafv2b  47261  funbrafv22b  47262
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