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Theorem mpt2fun 5545
Description: The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)
Hypothesis
Ref Expression
mpt2fun.1 𝐹 = (x A, y B𝐶)
Assertion
Ref Expression
mpt2fun Fun 𝐹
Distinct variable group:   x,y
Allowed substitution hints:   A(x,y)   B(x,y)   𝐶(x,y)   𝐹(x,y)

Proof of Theorem mpt2fun
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqtr3 2056 . . . . . 6 ((z = 𝐶 w = 𝐶) → z = w)
21ad2ant2l 477 . . . . 5 ((((x A y B) z = 𝐶) ((x A y B) w = 𝐶)) → z = w)
32gen2 1336 . . . 4 zw((((x A y B) z = 𝐶) ((x A y B) w = 𝐶)) → z = w)
4 eqeq1 2043 . . . . . 6 (z = w → (z = 𝐶w = 𝐶))
54anbi2d 437 . . . . 5 (z = w → (((x A y B) z = 𝐶) ↔ ((x A y B) w = 𝐶)))
65mo4 1958 . . . 4 (∃*z((x A y B) z = 𝐶) ↔ zw((((x A y B) z = 𝐶) ((x A y B) w = 𝐶)) → z = w))
73, 6mpbir 134 . . 3 ∃*z((x A y B) z = 𝐶)
87funoprab 5543 . 2 Fun {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
9 mpt2fun.1 . . . 4 𝐹 = (x A, y B𝐶)
10 df-mpt2 5460 . . . 4 (x A, y B𝐶) = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
119, 10eqtri 2057 . . 3 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
1211funeqi 4865 . 2 (Fun 𝐹 ↔ Fun {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)})
138, 12mpbir 134 1 Fun 𝐹
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   = wceq 1242   wcel 1390  ∃*wmo 1898  Fun wfun 4839  {coprab 5456  cmpt2 5457
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 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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-fun 4847  df-oprab 5459  df-mpt2 5460
This theorem is referenced by:  elmpt2cl  5640  ofexg  5658  mpt2exxg  5775  mpt2xopn0yelv  5795
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