Theorem fconst4 3851

Description: Two ways to express a constant function.

Assertion

Ref	Expression
fconst4	|- (F:A-->{B} <-> (F Fn A /\ (`'F"{B}) = A))

Proof of Theorem fconst4

Step	Hyp	Ref	Expression
1		fconst3 3850	. 2 |- (F:A-->{B} <-> (F Fn A /\ A (_ (`'F"{B})))
2		cnvimass 3423	. . . . . . 7 |- (`'F"{B}) (_ dom F
3	2	a1i 8	. . . . . 6 |- (F Fn A -> (`'F"{B}) (_ dom F)
4		fndm 3587	. . . . . 6 |- (F Fn A -> dom F = A)
5	3, 4	sseqtrd 2097	. . . . 5 |- (F Fn A -> (`'F"{B}) (_ A)
6	5	biantrurd 727	. . . 4 |- (F Fn A -> (A (_ (`'F"{B}) <-> ((`'F"{B}) (_ A /\ A (_ (`'F"{B}))))
7		eqss 2077	. . . 4 |- ((`'F"{B}) = A <-> ((`'F"{B}) (_ A /\ A (_ (`'F"{B})))
8	6, 7	syl6bbr 538	. . 3 |- (F Fn A -> (A (_ (`'F"{B}) <-> (`'F"{B}) = A))
9	8	pm5.32i 645	. 2 |- ((F Fn A /\ A (_ (`'F"{B})) <-> (F Fn A /\ (`'F"{B}) = A))
10	1, 9	bitr 173	1 |- (F:A-->{B} <-> (F Fn A /\ (`'F"{B}) = A))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   (_ wss 2047  {csn 2409  `'ccnv 3169  dom cdm 3170  "cima 3173   Fn wfn 3177  -->wf 3178

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198

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