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Theorem cnviinm 4801
Description: The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
Assertion
Ref Expression
cnviinm (y y A x A B = x A B)
Distinct variable groups:   x,A   y,A
Allowed substitution hints:   B(x,y)

Proof of Theorem cnviinm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2097 . . 3 (y = 𝑎 → (y A𝑎 A))
21cbvexv 1792 . 2 (y y A𝑎 𝑎 A)
3 eleq1 2097 . . . 4 (x = 𝑎 → (x A𝑎 A))
43cbvexv 1792 . . 3 (x x A𝑎 𝑎 A)
5 relcnv 4645 . . . 4 Rel x A B
6 r19.2m 3303 . . . . . . . 8 ((x x A x A B ⊆ (V × V)) → x A B ⊆ (V × V))
76expcom 109 . . . . . . 7 (x A B ⊆ (V × V) → (x x Ax A B ⊆ (V × V)))
8 relcnv 4645 . . . . . . . . 9 Rel B
9 df-rel 4294 . . . . . . . . 9 (Rel BB ⊆ (V × V))
108, 9mpbi 133 . . . . . . . 8 B ⊆ (V × V)
1110a1i 9 . . . . . . 7 (x AB ⊆ (V × V))
127, 11mprg 2372 . . . . . 6 (x x Ax A B ⊆ (V × V))
13 iinss 3698 . . . . . 6 (x A B ⊆ (V × V) → x A B ⊆ (V × V))
1412, 13syl 14 . . . . 5 (x x A x A B ⊆ (V × V))
15 df-rel 4294 . . . . 5 (Rel x A B x A B ⊆ (V × V))
1614, 15sylibr 137 . . . 4 (x x A → Rel x A B)
17 vex 2554 . . . . . . . 8 𝑏 V
18 vex 2554 . . . . . . . 8 𝑎 V
1917, 18opex 3956 . . . . . . 7 𝑏, 𝑎 V
20 eliin 3652 . . . . . . 7 (⟨𝑏, 𝑎 V → (⟨𝑏, 𝑎 x A Bx A𝑏, 𝑎 B))
2119, 20ax-mp 7 . . . . . 6 (⟨𝑏, 𝑎 x A Bx A𝑏, 𝑎 B)
2218, 17opelcnv 4459 . . . . . 6 (⟨𝑎, 𝑏 x A B ↔ ⟨𝑏, 𝑎 x A B)
2318, 17opex 3956 . . . . . . . 8 𝑎, 𝑏 V
24 eliin 3652 . . . . . . . 8 (⟨𝑎, 𝑏 V → (⟨𝑎, 𝑏 x A Bx A𝑎, 𝑏 B))
2523, 24ax-mp 7 . . . . . . 7 (⟨𝑎, 𝑏 x A Bx A𝑎, 𝑏 B)
2618, 17opelcnv 4459 . . . . . . . 8 (⟨𝑎, 𝑏 B ↔ ⟨𝑏, 𝑎 B)
2726ralbii 2324 . . . . . . 7 (x A𝑎, 𝑏 Bx A𝑏, 𝑎 B)
2825, 27bitri 173 . . . . . 6 (⟨𝑎, 𝑏 x A Bx A𝑏, 𝑎 B)
2921, 22, 283bitr4i 201 . . . . 5 (⟨𝑎, 𝑏 x A B ↔ ⟨𝑎, 𝑏 x A B)
3029eqrelriv 4375 . . . 4 ((Rel x A B Rel x A B) → x A B = x A B)
315, 16, 30sylancr 393 . . 3 (x x A x A B = x A B)
324, 31sylbir 125 . 2 (𝑎 𝑎 A x A B = x A B)
332, 32sylbi 114 1 (y y A x A B = x A B)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wex 1378   wcel 1390  wral 2300  wrex 2301  Vcvv 2551  wss 2911  cop 3369   ciin 3648   × cxp 4285  ccnv 4286  Rel wrel 4292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-iin 3650  df-br 3755  df-opab 3809  df-xp 4293  df-rel 4294  df-cnv 4295
This theorem is referenced by: (None)
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