Theorem mapval2 4335

Description: Alternate expression for the value of set exponentiation.

Hypotheses

Ref	Expression
elmap.1	|- A e. V
elmap.2	|- B e. V

Assertion

Ref	Expression
mapval2	|- (A ^m B) = (P~(B X. A) i^i {f | f Fn B})

Distinct variable group:   B,f

Proof of Theorem mapval2

Step	Hyp	Ref	Expression
1		ffn 3627	. . . . . 6 |- (g:B-->A -> g Fn B)
2		fssxp 3637	. . . . . 6 |- (g:B-->A -> g (_ (B X. A))
3	1, 2	jca 288	. . . . 5 |- (g:B-->A -> (g Fn B /\ g (_ (B X. A)))
4		rnss 3342	. . . . . . . 8 |- (g (_ (B X. A) -> ran g (_ ran ( B X. A))
5		rnxpss 3474	. . . . . . . . 9 |- ran ( B X. A) (_ A
6		sstr 2072	. . . . . . . . 9 |- ((ran g (_ ran ( B X. A) /\ ran ( B X. A) (_ A) -> ran g (_ A)
7	5, 6	mpan2 696	. . . . . . . 8 |- (ran g (_ ran ( B X. A) -> ran g (_ A)
8	4, 7	syl 10	. . . . . . 7 |- (g (_ (B X. A) -> ran g (_ A)
9	8	anim2i 335	. . . . . 6 |- ((g Fn B /\ g (_ (B X. A)) -> (g Fn B /\ ran g (_ A))
10		df-f 3194	. . . . . 6 |- (g:B-->A <-> (g Fn B /\ ran g (_ A))
11	9, 10	sylibr 200	. . . . 5 |- ((g Fn B /\ g (_ (B X. A)) -> g:B-->A)
12	3, 11	impbi 157	. . . 4 |- (g:B-->A <-> (g Fn B /\ g (_ (B X. A)))
13		ancom 435	. . . 4 |- ((g Fn B /\ g (_ (B X. A)) <-> (g (_ (B X. A) /\ g Fn B))
14	12, 13	bitr 173	. . 3 |- (g:B-->A <-> (g (_ (B X. A) /\ g Fn B))
15		elmap.1	. . . 4 |- A e. V
16		elmap.2	. . . 4 |- B e. V
17	15, 16	elmap 4334	. . 3 |- (g e. (A ^m B) <-> g:B-->A)
18		elin 2207	. . . 4 |- (g e. (P~(B X. A) i^i {f | f Fn B}) <-> (g e. P~(B X. A) /\ g e. {f | f Fn B}))
19		visset 1813	. . . . . 6 |- g e. V
20	19	elpw 2404	. . . . 5 |- (g e. P~(B X. A) <-> g (_ (B X. A))
21		fneq1 3582	. . . . . 6 |- (f = g -> (f Fn B <-> g Fn B))
22	19, 21	elab 1897	. . . . 5 |- (g e. {f | f Fn B} <-> g Fn B)
23	20, 22	anbi12i 482	. . . 4 |- ((g e. P~(B X. A) /\ g e. {f | f Fn B}) <-> (g (_ (B X. A) /\ g Fn B))
24	18, 23	bitr 173	. . 3 |- (g e. (P~(B X. A) i^i {f | f Fn B}) <-> (g (_ (B X. A) /\ g Fn B))
25	14, 17, 24	3bitr4 183	. 2 |- (g e. (A ^m B) <-> g e. (P~(B X. A) i^i {f | f Fn B}))
26	25	eqriv 1474	1 |- (A ^m B) = (P~(B X. A) i^i {f | f Fn B})

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   i^i cin 2046   (_ wss 2047  P~cpw 2401   X. cxp 3168  ran crn 3171   Fn wfn 3177  -->wf 3178  (class class class)co 3963   ^m cm 4322

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-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  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-sbc 1942  df-csb 2002  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-fv 3198  df-opr 3965  df-oprab 3966  df-map 4324

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