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Mirrors > Home > NFE Home > Th. List > phieq | GIF version |
Description: Equality law for the Phi operation. (Contributed by SF, 3-Feb-2015.) |
Ref | Expression |
---|---|
phieq | ⊢ (A = B → Phi A = Phi B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imakeq2 4225 | . 2 ⊢ (A = B → (((Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) “k A) = (((Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) “k B)) | |
2 | dfphi2 4569 | . 2 ⊢ Phi A = (((Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) “k A) | |
3 | dfphi2 4569 | . 2 ⊢ Phi B = (((Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) “k B) | |
4 | 1, 2, 3 | 3eqtr4g 2410 | 1 ⊢ (A = B → Phi A = Phi B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 Vcvv 2859 ∼ ccompl 3205 ∖ cdif 3206 ∪ cun 3207 ∩ cin 3208 ⊕ csymdif 3209 1cc1c 4134 ℘1cpw1 4135 ×k cxpk 4174 Ins2k cins2k 4176 Ins3k cins3k 4177 “k cimak 4179 SIk csik 4181 Imagekcimagek 4182 Sk cssetk 4183 Ik cidk 4184 Nn cnnc 4373 Phi cphi 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 4078 ax-sn 4087 |
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 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-opk 4058 df-1c 4136 df-pw1 4137 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-addc 4378 df-phi 4565 |
This theorem is referenced by: phi11 4596 proj1op 4600 proj2op 4601 phialllem1 4616 phialllem2 4617 opeq 4619 |
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