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Theorem opeq1 4579

Description: Equality theorem for ordered pairs. (Contributed by SF, 2-Jan-2015.)

**Assertion**
Ref	Expression
opeq1	⊢ (A = B → ⟨A, C⟩ = ⟨B, C⟩)

**Proof of Theorem opeq1**
Step	Hyp	Ref	Expression
1		imakeq2 4226	. . 3 ⊢ (A = B → (Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) “_k A) = (Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) “_k B))
2	1	uneq1d 3418	. 2 ⊢ (A = B → ((Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) “_k A) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c) “_k C)) = ((Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) “_k B) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c) “_k C)))
3		dfop2 4576	. 2 ⊢ ⟨A, C⟩ = ((Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) “_k A) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c) “_k C))
4		dfop2 4576	. 2 ⊢ ⟨B, C⟩ = ((Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) “_k B) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c) “_k C))
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → ⟨A, C⟩ = ⟨B, C⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1642 Vcvv 2860 ∼ ccompl 3206 ∖ cdif 3207 ∪ cun 3208 ∩ cin 3209 ⊕ csymdif 3210 {csn 3738 1_cc1c 4135 𝒫₁cpw1 4136 ×_k cxpk 4175 ^◡_kccnvk 4176 Ins2_k cins2k 4177 Ins3_k cins3k 4178 “_k cimak 4180 ∘_k ccomk 4181 SI_k csik 4182 Image_kcimagek 4183 S_k cssetk 4184 I_k cidk 4185 Nn cnnc 4374 0_cc0c 4375 ⟨cop 4562

This proof depends on 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 proof 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 used by: opeq12 4581 opeq1i 4582 opeq1d 4585 eqvinop 4607 cbvopab1 4633 cbvopab1s 4635 breq1 4643 opelopabsb 4698 el1st 4730 br1stg 4731 elswap 4741 dfima2 4746 dfco1 4749 dfsi2 4752 opeliunxp2 4823 ssrel 4845 ssopr 4847 br1st 4859 brswap2 4861 opabid2 4862 elsnres 4997 elxp4 5109 dfxp2 5114 fnunsn 5191 f1osng 5324 fvelrn 5414 fsng 5434 fvsng 5447 oveq1 5531 oprabid 5551 dfoprab2 5559 cbvoprab1 5568 oprabid2 5647 trtxp 5782 brtxp 5784 otelins2 5792 otelins3 5793 oqelins4 5795 qrpprod 5837 xpassen 6058 addccan2nclem1 6264

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