dfima2 - New Foundations Explorer

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Theorem dfima2 4746

Description: Express the image functor in terms of the set construction functions. (Contributed by SF, 7-Jan-2015.)

**Assertion**
Ref	Expression
dfima2	_k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k

Proof of Theorem dfima2 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ima 4728	. 2
2		vex 2863	. . . . 5
3	2	elimak 4260	. . . 4 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
4		setconslem6 4737	. . . . . . 7 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ $/\$
5		opeq1 4579	. . . . . . . 8
6	5	eleq1d 2419	. . . . . . 7
7		opeq2 4580	. . . . . . . 8
8	7	eleq1d 2419	. . . . . . 7
9		vex 2863	. . . . . . 7
10	4, 6, 8, 9, 2	opkelopkab 4247	. . . . . 6 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
11		df-br 4641	. . . . . 6
12	10, 11	bitr4i 243	. . . . 5 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
13	12	rexbii 2640	. . . 4 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
14	3, 13	bitri 240	. . 3 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k
15	14	abbi2i 2465	. 2 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k
16	1, 15	eqtr4i 2376	1 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k

Colors of variables: wff setvar class

Syntax hints:

wceq 1642

wcel 1710

cab 2339

wrex 2616

cvv 2860 ∼ ccompl 3206

cdif 3207

cun 3208

cin 3209

csymdif 3210

csn 3738

copk 4058 1_cc1c 4135

₁ cpw1 4136

_k cxpk 4175

_kccnvk 4176 Ins2_k cins2k 4177 Ins3_k cins3k 4178

_kcimak 4180

_k ccomk 4181 SI_k csik 4182 Image_kcimagek 4183 S_k cssetk 4184

_k cidk 4185 Nn cnnc 4374 0_cc0c 4375

cop 4562 class class class wbr 4640

cima 4723

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-addc 4379 df-nnc 4380 df-phi 4566 df-op 4567 df-br 4641 df-ima 4728

This theorem is referenced by: imaexg 4747

W3C validator