Theorem funssxp 3638

Description: Two ways of specifying a partial function from A to B.

Assertion

Ref	Expression
funssxp	|- ((Fun F /\ F (_ (A X. B)) <-> (F:dom F-->B /\ dom F (_ A))

Proof of Theorem funssxp

Step	Hyp	Ref	Expression
1		funfn 3542	. . . . . 6 |- (Fun F <-> F Fn dom F)
2	1	biimp 151	. . . . 5 |- (Fun F -> F Fn dom F)
3		rnss 3342	. . . . . 6 |- (F (_ (A X. B) -> ran F (_ ran ( A X. B))
4		rnxpss 3474	. . . . . . 7 |- ran ( A X. B) (_ B
5		sstr 2072	. . . . . . 7 |- ((ran F (_ ran ( A X. B) /\ ran ( A X. B) (_ B) -> ran F (_ B)
6	4, 5	mpan2 696	. . . . . 6 |- (ran F (_ ran ( A X. B) -> ran F (_ B)
7	3, 6	syl 10	. . . . 5 |- (F (_ (A X. B) -> ran F (_ B)
8	2, 7	anim12i 333	. . . 4 |- ((Fun F /\ F (_ (A X. B)) -> (F Fn dom F /\ ran F (_ B))
9		df-f 3194	. . . 4 |- (F:dom F-->B <-> (F Fn dom F /\ ran F (_ B))
10	8, 9	sylibr 200	. . 3 |- ((Fun F /\ F (_ (A X. B)) -> F:dom F-->B)
11		dmss 3310	. . . . 5 |- (F (_ (A X. B) -> dom F (_ dom ( A X. B))
12		dmxpss 3473	. . . . . 6 |- dom ( A X. B) (_ A
13		sstr 2072	. . . . . 6 |- ((dom F (_ dom ( A X. B) /\ dom ( A X. B) (_ A) -> dom F (_ A)
14	12, 13	mpan2 696	. . . . 5 |- (dom F (_ dom ( A X. B) -> dom F (_ A)
15	11, 14	syl 10	. . . 4 |- (F (_ (A X. B) -> dom F (_ A)
16	15	adantl 388	. . 3 |- ((Fun F /\ F (_ (A X. B)) -> dom F (_ A)
17	10, 16	jca 288	. 2 |- ((Fun F /\ F (_ (A X. B)) -> (F:dom F-->B /\ dom F (_ A))
18		ffun 3629	. . . 4 |- (F:dom F-->B -> Fun F)
19	18	adantr 389	. . 3 |- ((F:dom F-->B /\ dom F (_ A) -> Fun F)
20		fssxp 3637	. . . 4 |- (F:dom F-->B -> F (_ (dom F X. B))
21		ssid 2080	. . . . 5 |- B (_ B
22		ssxp 3256	. . . . 5 |- ((dom F (_ A /\ B (_ B) -> (dom F X. B) (_ (A X. B))
23	21, 22	mpan2 696	. . . 4 |- (dom F (_ A -> (dom F X. B) (_ (A X. B))
24	20, 23	sylan9ss 2075	. . 3 |- ((F:dom F-->B /\ dom F (_ A) -> F (_ (A X. B))
25	19, 24	jca 288	. 2 |- ((F:dom F-->B /\ dom F (_ A) -> (Fun F /\ F (_ (A X. B)))
26	17, 25	impbi 157	1 |- ((Fun F /\ F (_ (A X. B)) <-> (F:dom F-->B /\ dom F (_ A))

Syntax hints:   <-> wb 146   /\ wa 223   (_ wss 2047   X. cxp 3168  dom cdm 3170  ran crn 3171  Fun wfun 3176   Fn wfn 3177  -->wf 3178

This theorem is referenced by:  elpm2 4337

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-9 965  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-pow 2742  ax-pr 2779

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-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-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-f 3194

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