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Theorem cnviinm 4782
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 2078 . . 3 (y = 𝑎 → (y A𝑎 A))
21cbvexv 1773 . 2 (y y A𝑎 𝑎 A)
3 eleq1 2078 . . . 4 (x = 𝑎 → (x A𝑎 A))
43cbvexv 1773 . . 3 (x x A𝑎 𝑎 A)
5 relcnv 4626 . . . 4 Rel x A B
6 r19.2m 3282 . . . . . . . 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 4626 . . . . . . . . 9 Rel B
9 df-rel 4275 . . . . . . . . 9 (Rel BB ⊆ (V × V))
108, 9mpbi 133 . . . . . . . 8 B ⊆ (V × V)
1110a1i 9 . . . . . . 7 (x AB ⊆ (V × V))
127, 11mprg 2352 . . . . . 6 (x x Ax A B ⊆ (V × V))
13 iinss 3678 . . . . . 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 4275 . . . . 5 (Rel x A B x A B ⊆ (V × V))
1614, 15sylibr 137 . . . 4 (x x A → Rel x A B)
17 vex 2534 . . . . . . . 8 𝑏 V
18 vex 2534 . . . . . . . 8 𝑎 V
1917, 18opex 3936 . . . . . . 7 𝑏, 𝑎 V
20 eliin 3632 . . . . . . 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 4440 . . . . . 6 (⟨𝑎, 𝑏 x A B ↔ ⟨𝑏, 𝑎 x A B)
2318, 17opex 3936 . . . . . . . 8 𝑎, 𝑏 V
24 eliin 3632 . . . . . . . 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 4440 . . . . . . . 8 (⟨𝑎, 𝑏 B ↔ ⟨𝑏, 𝑎 B)
2726ralbii 2304 . . . . . . 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 4356 . . . 4 ((Rel x A B Rel x A B) → x A B = x A B)
315, 16, 30sylancr 395 . . 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 1226  wex 1358   wcel 1370  wral 2280  wrex 2281  Vcvv 2531  wss 2890  cop 3349   ciin 3628   × cxp 4266  ccnv 4267  Rel wrel 4273
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-iin 3630  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-cnv 4276
This theorem is referenced by: (None)
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