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Theorem mptfng 4965
 Description: The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)
Hypothesis
Ref Expression
mptfng.1 𝐹 = (x AB)
Assertion
Ref Expression
mptfng (x A B V ↔ 𝐹 Fn A)
Distinct variable group:   x,A
Allowed substitution hints:   B(x)   𝐹(x)

Proof of Theorem mptfng
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eueq 2706 . . 3 (B V ↔ ∃!y y = B)
21ralbii 2324 . 2 (x A B V ↔ x A ∃!y y = B)
3 mptfng.1 . . . 4 𝐹 = (x AB)
4 df-mpt 3810 . . . 4 (x AB) = {⟨x, y⟩ ∣ (x A y = B)}
53, 4eqtri 2057 . . 3 𝐹 = {⟨x, y⟩ ∣ (x A y = B)}
65fnopabg 4963 . 2 (x A ∃!y y = B𝐹 Fn A)
72, 6bitri 173 1 (x A B V ↔ 𝐹 Fn A)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∃!weu 1897  ∀wral 2300  Vcvv 2551  {copab 3807   ↦ cmpt 3808   Fn wfn 4839 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 3865  ax-pow 3917  ax-pr 3934 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-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-mpt 3810  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-fun 4846  df-fn 4847 This theorem is referenced by:  fnmpt  4966  fnmpti  4968  mpteqb  5202
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