Theorem reldmmap 8841

Description: Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Assertion

Ref	Expression
reldmmap	⊢ Rel dom ↑m

Proof of Theorem reldmmap

Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-map 8835	. 2 ⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
2	1	reldmmpo 7557	1 ⊢ Rel dom ↑m

This proof depends on syntax axioms:  {cab 2744  Vcvv 3458  dom cdm 5666  Rel wrel 5671  ⟶wf 6539   ↑_m cmap 8833

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-dm 5676  df-oprab 7427  df-mpo 7428  df-map 8835

This theorem is used by:  mapssfset  8857  mapdom2  9146  efmndbas  18961  smatrcl  34217  mapco2g  43486  naryfvalixp  49450  1aryenef  49466  2aryenef  49477