Theorem dff4 3822

Description: Alternate definition of a mapping.

Assertion

Ref	Expression
dff4	|- (F:A-->B <-> (F Fn A /\ F (_ (A X. B)))

Proof of Theorem dff4

Step	Hyp	Ref	Expression
1		ffn 3633	. . 3 |- (F:A-->B -> F Fn A)
2		fssxp 3643	. . 3 |- (F:A-->B -> F (_ (A X. B))
3	1, 2	jca 288	. 2 |- (F:A-->B -> (F Fn A /\ F (_ (A X. B)))
4		rnss 3348	. . . . 5 |- (F (_ (A X. B) -> ran F (_ ran ( A X. B))
5		rnxpss 3480	. . . . . 6 |- ran ( A X. B) (_ B
6		sstr 2075	. . . . . 6 |- ((ran F (_ ran ( A X. B) /\ ran ( A X. B) (_ B) -> ran F (_ B)
7	5, 6	mpan2 698	. . . . 5 |- (ran F (_ ran ( A X. B) -> ran F (_ B)
8	4, 7	syl 10	. . . 4 |- (F (_ (A X. B) -> ran F (_ B)
9	8	anim2i 335	. . 3 |- ((F Fn A /\ F (_ (A X. B)) -> (F Fn A /\ ran F (_ B))
10		df-f 3200	. . 3 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
11	9, 10	sylibr 200	. 2 |- ((F Fn A /\ F (_ (A X. B)) -> F:A-->B)
12	3, 11	impbi 157	1 |- (F:A-->B <-> (F Fn A /\ F (_ (A X. B)))

Syntax hints:   <-> wb 146   /\ wa 223   (_ wss 2050   X. cxp 3174  ran crn 3177   Fn wfn 3183  -->wf 3184

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195  df-fun 3198  df-fn 3199  df-f 3200

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