Theorem fconst3 3841

Description: Two ways to express a constant function.

Assertion

Ref	Expression
fconst3	|- (F:A-->{B} <-> (F Fn A /\ A (_ (`'F"{B})))

Proof of Theorem fconst3

Step	Hyp	Ref	Expression
1		fconstfv 3840	. 2 |- (F:A-->{B} <-> (F Fn A /\ A.x e. A (F` x) = B))
2		funconstss 3799	. . . 4 |- ((Fun F /\ A (_ dom F) -> (A.x e. A (F` x) = B <-> A (_ (`'F"{B})))
3		fnfun 3577	. . . 4 |- (F Fn A -> Fun F)
4		fndm 3579	. . . . 5 |- (F Fn A -> dom F = A)
5		eqimss2 2106	. . . . 5 |- (dom F = A -> A (_ dom F)
6	4, 5	syl 10	. . . 4 |- (F Fn A -> A (_ dom F)
7	2, 3, 6	sylanc 471	. . 3 |- (F Fn A -> (A.x e. A (F` x) = B <-> A (_ (`'F"{B})))
8	7	pm5.32i 644	. 2 |- ((F Fn A /\ A.x e. A (F` x) = B) <-> (F Fn A /\ A (_ (`'F"{B})))
9	1, 8	bitr 173	1 |- (F:A-->{B} <-> (F Fn A /\ A (_ (`'F"{B})))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 954  A.wral 1642   (_ wss 2043  {csn 2405  `'ccnv 3164  dom cdm 3165  "cima 3168  Fun wfun 3171   Fn wfn 3172  -->wf 3173  ` cfv 3177

This theorem is referenced by:  fconst4 3842  dnsconst 7738

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fo 3191  df-fv 3193

Copyright terms: Public domain