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Theorem opexg 4588

Description: An ordered pair of two sets is a set. (Contributed by SF, 2-Jan-2015.)

**Assertion**
Ref	Expression
opexg	$/\$

**Proof of Theorem opexg**
Step	Hyp	Ref	Expression
1		dfop2 4576	. 2 Image_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 ∼ 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_k
2		addcexlem 4383	. . . . . . . . 9 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
3		1cex 4143	. . . . . . . . . . 11 1_c
4	3	pw1ex 4304	. . . . . . . . . 10 ₁ 1_c
5	4	pw1ex 4304	. . . . . . . . 9 ₁ ₁ 1_c
6	2, 5	imakex 4301	. . . . . . . 8 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
7	6	imagekex 4313	. . . . . . 7 Image_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
8		nncex 4397	. . . . . . . 8 Nn
9		vvex 4110	. . . . . . . 8
10	8, 9	xpkex 4290	. . . . . . 7 Nn _k
11	7, 10	inex 4106	. . . . . 6 Image_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
12		idkex 4315	. . . . . . 7 _k
13	8	complex 4105	. . . . . . . 8 ∼ Nn
14	13, 9	xpkex 4290	. . . . . . 7 ∼ Nn _k
15	12, 14	inex 4106	. . . . . 6 _k ∼ Nn _k
16	11, 15	unex 4107	. . . . 5 Image_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
17	16	imagekex 4313	. . . 4 Image_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
18		imakexg 4300	. . . 4 Image_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 $/\$ Image_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
19	17, 18	mpan 651	. . 3 Image_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
20		ssetkex 4295	. . . . . . . 8 S_k
21	20	ins2kex 4308	. . . . . . 7 Ins2_k S_k
22	17	cnvkex 4288	. . . . . . . . . 10 _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
23	22, 20	cokex 4311	. . . . . . . . 9 _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
24		snex 4112	. . . . . . . . . 10 0_c
25	24, 9	xpkex 4290	. . . . . . . . 9 0_c _k
26	23, 25	unex 4107	. . . . . . . 8 _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
27	26	ins3kex 4309	. . . . . . 7 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
28	21, 27	symdifex 4109	. . . . . 6 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
29	28, 5	imakex 4301	. . . . 5 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
30	29	complex 4105	. . . 4 ∼ 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
31		imakexg 4300	. . . 4 ∼ 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 $/\$ ∼ 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_k
32	30, 31	mpan 651	. . 3 ∼ 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_k
33		unexg 4102	. . 3 Image_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 $/\$ ∼ 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_k Image_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 ∼ 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_k
34	19, 32, 33	syl2an 463	. 2 $/\$ Image_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 ∼ 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_k
35	1, 34	syl5eqel 2437	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 358

wcel 1710

cvv 2860 ∼ ccompl 3206

cdif 3207

cun 3208

cin 3209

csymdif 3210

csn 3738 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

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

This theorem is referenced by: opex 4589 opexb 4604 opeliunxp 4821 opbrop 4842 eloprabga 5579 brcupg 5815 brcomposeg 5820 brcrossg 5849 dmfrec 6317 fnfreclem2 6319 fnfreclem3 6320 frec0 6322

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