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Theorem eufnfv 5310
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 5308 . . . 4 (x AB) V
3 eqeq2 2027 . . . . . 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 1683 . . . 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 2620 . . 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 2018 . . . . . . 7 (x AB) = (x AB)
97, 8fnmpti 4949 . . . . . 6 (x AB) Fn A
10 fneq1 4909 . . . . . 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 5140 . . . . . . 7 (f Fn Af = (x A ↦ (fx)))
1413eqeq1d 2026 . . . . . 6 (f Fn A → (f = (x AB) ↔ (x A ↦ (fx)) = (x AB)))
15 funfvex 5113 . . . . . . . . 9 ((Fun f x dom f) → (fx) V)
1615funfni 4921 . . . . . . . 8 ((f Fn A x A) → (fx) V)
1716ralrimiva 2366 . . . . . . 7 (f Fn Ax A (fx) V)
18 mpteqb 5182 . . . . . . 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 430 . . . 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 1316 . 2 yf((f Fn A x A (fx) = B) ↔ f = y)
24 df-eu 1881 . 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 1224   = wceq 1226  wex 1358   wcel 1370  ∃!weu 1878  wral 2280  Vcvv 2531  cmpt 3788   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-coll 3842  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-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  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-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833
This theorem is referenced by: (None)
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