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Theorem funbrafv22b 45892
Description: Equivalence of function value and binary relation, analogous to funbrfvb 6942. (Contributed by AV, 6-Sep-2022.)
Assertion
Ref Expression
funbrafv22b ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))

Proof of Theorem funbrafv22b
StepHypRef Expression
1 funfn 6574 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fnbrafv2b 45890 . 2 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))
31, 2sylanb 582 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   class class class wbr 5146  dom cdm 5674  Fun wfun 6533   Fn wfn 6534  ''''cafv2 45850
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-res 5686  df-iota 6491  df-fun 6541  df-fn 6542  df-dfat 45761  df-afv2 45851
This theorem is referenced by: (None)
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