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Theorem eufnfv 5332
 Description: A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
Hypotheses
Ref Expression
eufnfv.1 A V
eufnfv.2 B V
Assertion
Ref Expression
eufnfv ∃!f(f Fn A x A (fx) = B)
Distinct variable groups:   x,f,A   B,f
Allowed substitution hint:   B(x)

Proof of Theorem eufnfv
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eufnfv.1 . . . . 5 A V
21mptex 5330 . . . 4 (x AB) V
3 eqeq2 2046 . . . . . 6 (y = (x AB) → (f = yf = (x AB)))
43bibi2d 221 . . . . 5 (y = (x AB) → (((f Fn A x A (fx) = B) ↔ f = y) ↔ ((f Fn A x A (fx) = B) ↔ f = (x AB))))
54albidv 1702 . . . 4 (y = (x AB) → (f((f Fn A x A (fx) = B) ↔ f = y) ↔ f((f Fn A x A (fx) = B) ↔ f = (x AB))))
62, 5spcev 2641 . . 3 (f((f Fn A x A (fx) = B) ↔ f = (x AB)) → yf((f Fn A x A (fx) = B) ↔ f = y))
7 eufnfv.2 . . . . . . 7 B V
8 eqid 2037 . . . . . . 7 (x AB) = (x AB)
97, 8fnmpti 4970 . . . . . 6 (x AB) Fn A
10 fneq1 4930 . . . . . 6 (f = (x AB) → (f Fn A ↔ (x AB) Fn A))
119, 10mpbiri 157 . . . . 5 (f = (x AB) → f Fn A)
1211pm4.71ri 372 . . . 4 (f = (x AB) ↔ (f Fn A f = (x AB)))
13 dffn5im 5162 . . . . . . 7 (f Fn Af = (x A ↦ (fx)))
1413eqeq1d 2045 . . . . . 6 (f Fn A → (f = (x AB) ↔ (x A ↦ (fx)) = (x AB)))
15 funfvex 5135 . . . . . . . . 9 ((Fun f x dom f) → (fx) V)
1615funfni 4942 . . . . . . . 8 ((f Fn A x A) → (fx) V)
1716ralrimiva 2386 . . . . . . 7 (f Fn Ax A (fx) V)
18 mpteqb 5204 . . . . . . 7 (x A (fx) V → ((x A ↦ (fx)) = (x AB) ↔ x A (fx) = B))
1917, 18syl 14 . . . . . 6 (f Fn A → ((x A ↦ (fx)) = (x AB) ↔ x A (fx) = B))
2014, 19bitrd 177 . . . . 5 (f Fn A → (f = (x AB) ↔ x A (fx) = B))
2120pm5.32i 427 . . . 4 ((f Fn A f = (x AB)) ↔ (f Fn A x A (fx) = B))
2212, 21bitr2i 174 . . 3 ((f Fn A x A (fx) = B) ↔ f = (x AB))
236, 22mpg 1337 . 2 yf((f Fn A x A (fx) = B) ↔ f = y)
24 df-eu 1900 . 2 (∃!f(f Fn A x A (fx) = B) ↔ yf((f Fn A x A (fx) = B) ↔ f = y))
2523, 24mpbir 134 1 ∃!f(f Fn A x A (fx) = B)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃!weu 1897  ∀wral 2300  Vcvv 2551   ↦ cmpt 3809   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-coll 3863  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-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  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-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853 This theorem is referenced by: (None)
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