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Theorem funbrfv 5155
Description: The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
funbrfv (Fun 𝐹 → (A𝐹B → (𝐹A) = B))

Proof of Theorem funbrfv
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 funrel 4862 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 brrelex2 4326 . . . 4 ((Rel 𝐹 A𝐹B) → B V)
31, 2sylan 267 . . 3 ((Fun 𝐹 A𝐹B) → B V)
4 breq2 3759 . . . . . 6 (y = B → (A𝐹yA𝐹B))
54anbi2d 437 . . . . 5 (y = B → ((Fun 𝐹 A𝐹y) ↔ (Fun 𝐹 A𝐹B)))
6 eqeq2 2046 . . . . 5 (y = B → ((𝐹A) = y ↔ (𝐹A) = B))
75, 6imbi12d 223 . . . 4 (y = B → (((Fun 𝐹 A𝐹y) → (𝐹A) = y) ↔ ((Fun 𝐹 A𝐹B) → (𝐹A) = B)))
8 funeu 4869 . . . . . 6 ((Fun 𝐹 A𝐹y) → ∃!y A𝐹y)
9 tz6.12-1 5143 . . . . . 6 ((A𝐹y ∃!y A𝐹y) → (𝐹A) = y)
108, 9sylan2 270 . . . . 5 ((A𝐹y (Fun 𝐹 A𝐹y)) → (𝐹A) = y)
1110anabss7 517 . . . 4 ((Fun 𝐹 A𝐹y) → (𝐹A) = y)
127, 11vtoclg 2607 . . 3 (B V → ((Fun 𝐹 A𝐹B) → (𝐹A) = B))
133, 12mpcom 32 . 2 ((Fun 𝐹 A𝐹B) → (𝐹A) = B)
1413ex 108 1 (Fun 𝐹 → (A𝐹B → (𝐹A) = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  ∃!weu 1897  Vcvv 2551   class class class wbr 3755  Rel wrel 4293  Fun wfun 4839  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853
This theorem is referenced by:  funopfv  5156  fnbrfvb  5157  fvelima  5168  fvi  5173  fmptco  5273  fliftfun  5379  fliftval  5383  tfrlem5  5871
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