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Theorem fnbrfvb 5157
Description: Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fnbrfvb ((𝐹 Fn A B A) → ((𝐹B) = 𝐶B𝐹𝐶))

Proof of Theorem fnbrfvb
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eqid 2037 . . . 4 (𝐹B) = (𝐹B)
2 funfvex 5135 . . . . . 6 ((Fun 𝐹 B dom 𝐹) → (𝐹B) V)
32funfni 4942 . . . . 5 ((𝐹 Fn A B A) → (𝐹B) V)
4 eqeq2 2046 . . . . . . . 8 (x = (𝐹B) → ((𝐹B) = x ↔ (𝐹B) = (𝐹B)))
5 breq2 3759 . . . . . . . 8 (x = (𝐹B) → (B𝐹xB𝐹(𝐹B)))
64, 5bibi12d 224 . . . . . . 7 (x = (𝐹B) → (((𝐹B) = xB𝐹x) ↔ ((𝐹B) = (𝐹B) ↔ B𝐹(𝐹B))))
76imbi2d 219 . . . . . 6 (x = (𝐹B) → (((𝐹 Fn A B A) → ((𝐹B) = xB𝐹x)) ↔ ((𝐹 Fn A B A) → ((𝐹B) = (𝐹B) ↔ B𝐹(𝐹B)))))
8 fneu 4946 . . . . . . 7 ((𝐹 Fn A B A) → ∃!x B𝐹x)
9 tz6.12c 5146 . . . . . . 7 (∃!x B𝐹x → ((𝐹B) = xB𝐹x))
108, 9syl 14 . . . . . 6 ((𝐹 Fn A B A) → ((𝐹B) = xB𝐹x))
117, 10vtoclg 2607 . . . . 5 ((𝐹B) V → ((𝐹 Fn A B A) → ((𝐹B) = (𝐹B) ↔ B𝐹(𝐹B))))
123, 11mpcom 32 . . . 4 ((𝐹 Fn A B A) → ((𝐹B) = (𝐹B) ↔ B𝐹(𝐹B)))
131, 12mpbii 136 . . 3 ((𝐹 Fn A B A) → B𝐹(𝐹B))
14 breq2 3759 . . 3 ((𝐹B) = 𝐶 → (B𝐹(𝐹B) ↔ B𝐹𝐶))
1513, 14syl5ibcom 144 . 2 ((𝐹 Fn A B A) → ((𝐹B) = 𝐶B𝐹𝐶))
16 fnfun 4939 . . . 4 (𝐹 Fn A → Fun 𝐹)
17 funbrfv 5155 . . . 4 (Fun 𝐹 → (B𝐹𝐶 → (𝐹B) = 𝐶))
1816, 17syl 14 . . 3 (𝐹 Fn A → (B𝐹𝐶 → (𝐹B) = 𝐶))
1918adantr 261 . 2 ((𝐹 Fn A B A) → (B𝐹𝐶 → (𝐹B) = 𝐶))
2015, 19impbid 120 1 ((𝐹 Fn A B A) → ((𝐹B) = 𝐶B𝐹𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  ∃!weu 1897  Vcvv 2551   class class class wbr 3755  Fun wfun 4839   Fn wfn 4840  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-bndl 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-fn 4848  df-fv 4853
This theorem is referenced by:  fnopfvb  5158  funbrfvb  5159  dffn5im  5162  fnsnfv  5175  fndmdif  5215  dffo4  5258  dff13  5350  isoini  5400  1stconst  5784  2ndconst  5785
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