Theorem mapval2 4341

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 3633	. . . . . 6 |- (g:B-->A -> g Fn B)
2		fssxp 3643	. . . . . 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 3348	. . . . . . . 8 |- (g (_ (B X. A) -> ran g (_ ran ( B X. A))
5		rnxpss 3480	. . . . . . . . 9 |- ran ( B X. A) (_ A
6		sstr 2075	. . . . . . . . 9 |- ((ran g (_ ran ( B X. A) /\ ran ( B X. A) (_ A) -> ran g (_ A)
7	5, 6	mpan2 698	. . . . . . . 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 3200	. . . . . 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 437	. . . 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 4340	. . 3 |- (g e. (A ^m B) <-> g:B-->A)
18		elin 2210	. . . 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 1816	. . . . . 6 |- g e. V
20	19	elpw 2408	. . . . 5 |- (g e. P~(B X. A) <-> g (_ (B X. A))
21		fneq1 3588	. . . . . 6 |- (f = g -> (f Fn B <-> g Fn B))
22	19, 21	elab 1900	. . . . 5 |- (g e. {f | f Fn B} <-> g Fn B)
23	20, 22	anbi12i 484	. . . 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 1477	1 |- (A ^m B) = (P~(B X. A) i^i {f | f Fn B})

Syntax hints:   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466  Vcvv 1814   i^i cin 2049   (_ wss 2050  P~cpw 2405   X. cxp 3174  ran crn 3177   Fn wfn 3183  -->wf 3184  (class class class)co 3969   ^m cm 4328

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-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  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-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  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-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-oprab 3972  df-map 4330

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