Theorem mapenlem1 4489

Description: Lemma for mapen 4491.

Hypotheses

Ref	Expression
mapenlem.1	|- A e. V
mapenlem.2	|- B e. V
mapenlem.3	|- C e. V
mapenlem.4	|- D e. V
mapenlem.5	|- H = {<.x, y>. | (x e. (A ^m C) /\ y = ((f o. x) o. `'g))}

Assertion

Ref	Expression
mapenlem1	|- ((((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ z:C-->A) /\ v e. C) -> ((H` z)` (g` v)) = (f` (z` v)))

Distinct variable groups:   f,g,x,y,z,v,A   B,f,g,x,y,z,v   C,f,g,x,y,z,v   D,f,g,x,y,z,v   z,H,v

Proof of Theorem mapenlem1

Step	Hyp	Ref	Expression
1		mapenlem.1	. . . . . 6 |- A e. V
2		mapenlem.3	. . . . . 6 |- C e. V
3	1, 2	elmap 4334	. . . . 5 |- (z e. (A ^m C) <-> z:C-->A)
4		coeq2 3282	. . . . . . 7 |- (x = z -> (f o. x) = (f o. z))
5	4	coeq1d 3285	. . . . . 6 |- (x = z -> ((f o. x) o. `'g) = ((f o. z) o. `'g))
6		mapenlem.5	. . . . . 6 |- H = {<.x, y>. | (x e. (A ^m C) /\ y = ((f o. x) o. `'g))}
7		visset 1813	. . . . . . . 8 |- f e. V
8		visset 1813	. . . . . . . 8 |- z e. V
9	7, 8	coex 3525	. . . . . . 7 |- (f o. z) e. V
10		visset 1813	. . . . . . . 8 |- g e. V
11	10	cnvex 3520	. . . . . . 7 |- `'g e. V
12	9, 11	coex 3525	. . . . . 6 |- ((f o. z) o. `'g) e. V
13	5, 6, 12	fvopab4 3780	. . . . 5 |- (z e. (A ^m C) -> (H` z) = ((f o. z) o. `'g))
14	3, 13	sylbir 201	. . . 4 |- (z:C-->A -> (H` z) = ((f o. z) o. `'g))
15	14	fveq1d 3726	. . 3 |- (z:C-->A -> ((H` z)` (g` v)) = (((f o. z) o. `'g)` (g` v)))
16	15	ad2antlr 405	. 2 |- ((((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ z:C-->A) /\ v e. C) -> ((H` z)` (g` v)) = (((f o. z) o. `'g)` (g` v)))
17		f1ococnv1 3709	. . . . . . . . . 10 |- (g:C-1-1-onto->D -> (`'g o. g) = (I |` C))
18	17	coeq2d 3286	. . . . . . . . 9 |- (g:C-1-1-onto->D -> ((f o. z) o. (`'g o. g)) = ((f o. z) o. (I |` C)))
19		fcoi1 3645	. . . . . . . . 9 |- ((f o. z):C-->B -> ((f o. z) o. (I |` C)) = (f o. z))
20	18, 19	sylan9eqr 1529	. . . . . . . 8 |- (((f o. z):C-->B /\ g:C-1-1-onto->D) -> ((f o. z) o. (`'g o. g)) = (f o. z))
21		fco 3636	. . . . . . . . 9 |- ((f:A-->B /\ z:C-->A) -> (f o. z):C-->B)
22		f1of 3689	. . . . . . . . 9 |- (f:A-1-1-onto->B -> f:A-->B)
23	21, 22	sylan 448	. . . . . . . 8 |- ((f:A-1-1-onto->B /\ z:C-->A) -> (f o. z):C-->B)
24	20, 23	sylan 448	. . . . . . 7 |- (((f:A-1-1-onto->B /\ z:C-->A) /\ g:C-1-1-onto->D) -> ((f o. z) o. (`'g o. g)) = (f o. z))
25	24	an1rs 489	. . . . . 6 |- (((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ z:C-->A) -> ((f o. z) o. (`'g o. g)) = (f o. z))
26		coass 3512	. . . . . 6 |- (((f o. z) o. `'g) o. g) = ((f o. z) o. (`'g o. g))
27	25, 26	syl5eq 1519	. . . . 5 |- (((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ z:C-->A) -> (((f o. z) o. `'g) o. g) = (f o. z))
28	27	fveq1d 3726	. . . 4 |- (((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ z:C-->A) -> ((((f o. z) o. `'g) o. g)` v) = ((f o. z)` v))
29	28	adantr 389	. . 3 |- ((((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ z:C-->A) /\ v e. C) -> ((((f o. z) o. `'g) o. g)` v) = ((f o. z)` v))
30		fvco3 3776	. . . . 5 |- ((Fun ((f o. z) o. `'g) /\ g:C-->D /\ v e. C) -> ((((f o. z) o. `'g) o. g)` v) = (((f o. z) o. `'g)` (g` v)))
31	30	3expa 833	. . . 4 |- (((Fun ((f o. z) o. `'g) /\ g:C-->D) /\ v e. C) -> ((((f o. z) o. `'g) o. g)` v) = (((f o. z) o. `'g)` (g` v)))
32		funco 3550	. . . . . . . 8 |- ((Fun (f o. z) /\ Fun `'g) -> Fun ((f o. z) o. `'g))
33		funco 3550	. . . . . . . . 9 |- ((Fun f /\ Fun z) -> Fun (f o. z))
34		f1ofun 3691	. . . . . . . . 9 |- (f:A-1-1-onto->B -> Fun f)
35		ffun 3629	. . . . . . . . 9 |- (z:C-->A -> Fun z)
36	33, 34, 35	syl2an 454	. . . . . . . 8 |- ((f:A-1-1-onto->B /\ z:C-->A) -> Fun (f o. z))
37		f1o3 3694	. . . . . . . . 9 |- (g:C-1-1-onto->D <-> (g:C-onto->D /\ Fun `'g))
38	37	pm3.27bi 326	. . . . . . . 8 |- (g:C-1-1-onto->D -> Fun `'g)
39	32, 36, 38	syl2an 454	. . . . . . 7 |- (((f:A-1-1-onto->B /\ z:C-->A) /\ g:C-1-1-onto->D) -> Fun ((f o. z) o. `'g))
40		f1of 3689	. . . . . . 7 |- (g:C-1-1-onto->D -> g:C-->D)
41	39, 40	anim12i 333	. . . . . 6 |- ((((f:A-1-1-onto->B /\ z:C-->A) /\ g:C-1-1-onto->D) /\ g:C-1-1-onto->D) -> (Fun ((f o. z) o. `'g) /\ g:C-->D))
42	41	anabss3 500	. . . . 5 |- (((f:A-1-1-onto->B /\ z:C-->A) /\ g:C-1-1-onto->D) -> (Fun ((f o. z) o. `'g) /\ g:C-->D))
43	42	an1rs 489	. . . 4 |- (((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ z:C-->A) -> (Fun ((f o. z) o. `'g) /\ g:C-->D))
44	31, 43	sylan 448	. . 3 |- ((((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ z:C-->A) /\ v e. C) -> ((((f o. z) o. `'g) o. g)` v) = (((f o. z) o. `'g)` (g` v)))
45		fvco3 3776	. . . . . . 7 |- ((Fun f /\ z:C-->A /\ v e. C) -> ((f o. z)` v) = (f` (z` v)))
46	45	3expb 834	. . . . . 6 |- ((Fun f /\ (z:C-->A /\ v e. C)) -> ((f o. z)` v) = (f` (z` v)))
47	46, 34	sylan 448	. . . . 5 |- ((f:A-1-1-onto->B /\ (z:C-->A /\ v e. C)) -> ((f o. z)` v) = (f` (z` v)))
48	47	adantlr 393	. . . 4 |- (((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ (z:C-->A /\ v e. C)) -> ((f o. z)` v) = (f` (z` v)))
49	48	anassrs 441	. . 3 |- ((((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ z:C-->A) /\ v e. C) -> ((f o. z)` v) = (f` (z` v)))
50	29, 44, 49	3eqtr3d 1515	. 2 |- ((((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ z:C-->A) /\ v e. C) -> (((f o. z) o. `'g)` (g` v)) = (f` (z` v)))
51	16, 50	eqtrd 1507	1 |- ((((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ z:C-->A) /\ v e. C) -> ((H` z)` (g` v)) = (f` (z` v)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  {copab 2666  Icid 2831  `'ccnv 3169   |` cres 3172   o. ccom 3174  Fun wfun 3176  -->wf 3178  -onto->wfo 3180  -1-1-onto->wf1o 3181  ` cfv 3182  (class class class)co 3963   ^m cm 4322

This theorem is referenced by:  mapenlem2 4490

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-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-map 4324

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