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Theorem fnbrfvb 5135
 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 2018 . . . 4 (𝐹B) = (𝐹B)
2 funfvex 5113 . . . . . 6 ((Fun 𝐹 B dom 𝐹) → (𝐹B) V)
32funfni 4921 . . . . 5 ((𝐹 Fn A B A) → (𝐹B) V)
4 eqeq2 2027 . . . . . . . 8 (x = (𝐹B) → ((𝐹B) = x ↔ (𝐹B) = (𝐹B)))
5 breq2 3738 . . . . . . . 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 4925 . . . . . . 7 ((𝐹 Fn A B A) → ∃!x B𝐹x)
9 tz6.12c 5124 . . . . . . 7 (∃!x B𝐹x → ((𝐹B) = xB𝐹x))
108, 9syl 14 . . . . . 6 ((𝐹 Fn A B A) → ((𝐹B) = xB𝐹x))
117, 10vtoclg 2586 . . . . 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 3738 . . 3 ((𝐹B) = 𝐶 → (B𝐹(𝐹B) ↔ B𝐹𝐶))
1513, 14syl5ibcom 144 . 2 ((𝐹 Fn A B A) → ((𝐹B) = 𝐶B𝐹𝐶))
16 fnfun 4918 . . . 4 (𝐹 Fn A → Fun 𝐹)
17 funbrfv 5133 . . . 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 1226   ∈ wcel 1370  ∃!weu 1878  Vcvv 2531   class class class wbr 3734  Fun wfun 4819   Fn wfn 4820  ‘cfv 4825 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833 This theorem is referenced by:  fnopfvb  5136  funbrfvb  5137  dffn5im  5140  fnsnfv  5153  fndmdif  5193  dffo4  5236  dff13  5328  isoini  5378  1stconst  5761  2ndconst  5762
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