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Theorem mpt2fun 5522
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 2037 . . . . . 6 ((z = 𝐶 w = 𝐶) → z = w)
21ad2ant2l 465 . . . . 5 ((((x A y B) z = 𝐶) ((x A y B) w = 𝐶)) → z = w)
32gen2 1315 . . . 4 zw((((x A y B) z = 𝐶) ((x A y B) w = 𝐶)) → z = w)
4 eqeq1 2024 . . . . . 6 (z = w → (z = 𝐶w = 𝐶))
54anbi2d 440 . . . . 5 (z = w → (((x A y B) z = 𝐶) ↔ ((x A y B) w = 𝐶)))
65mo4 1939 . . . 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 5520 . 2 Fun {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
9 mpt2fun.1 . . . 4 𝐹 = (x A, y B𝐶)
10 df-mpt2 5437 . . . 4 (x A, y B𝐶) = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
119, 10eqtri 2038 . . 3 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
1211funeqi 4844 . 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 1224   = wceq 1226   wcel 1370  ∃*wmo 1879  Fun wfun 4819  {coprab 5433  cmpt2 5434
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-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-fun 4827  df-oprab 5436  df-mpt2 5437
This theorem is referenced by:  elmpt2cl  5617  ofexg  5635  mpt2exxg  5752  mpt2xopn0yelv  5772
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