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Theorem cnviin 6119
 Description: The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
cnviin (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem cnviin
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5943 . 2 Rel 𝑥𝐴 𝐵
2 relcnv 5943 . . . . . . 7 Rel 𝐵
3 df-rel 5534 . . . . . . 7 (Rel 𝐵𝐵 ⊆ (V × V))
42, 3mpbi 233 . . . . . 6 𝐵 ⊆ (V × V)
54rgenw 3082 . . . . 5 𝑥𝐴 𝐵 ⊆ (V × V)
6 r19.2z 4391 . . . . 5 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ⊆ (V × V)) → ∃𝑥𝐴 𝐵 ⊆ (V × V))
75, 6mpan2 690 . . . 4 (𝐴 ≠ ∅ → ∃𝑥𝐴 𝐵 ⊆ (V × V))
8 iinss 4948 . . . 4 (∃𝑥𝐴 𝐵 ⊆ (V × V) → 𝑥𝐴 𝐵 ⊆ (V × V))
97, 8syl 17 . . 3 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 ⊆ (V × V))
10 df-rel 5534 . . 3 (Rel 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ (V × V))
119, 10sylibr 237 . 2 (𝐴 ≠ ∅ → Rel 𝑥𝐴 𝐵)
12 opex 5327 . . . . 5 𝑏, 𝑎⟩ ∈ V
13 eliin 4891 . . . . 5 (⟨𝑏, 𝑎⟩ ∈ V → (⟨𝑏, 𝑎⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵))
1412, 13ax-mp 5 . . . 4 (⟨𝑏, 𝑎⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵)
15 vex 3413 . . . . 5 𝑎 ∈ V
16 vex 3413 . . . . 5 𝑏 ∈ V
1715, 16opelcnv 5726 . . . 4 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑥𝐴 𝐵)
18 opex 5327 . . . . . 6 𝑎, 𝑏⟩ ∈ V
19 eliin 4891 . . . . . 6 (⟨𝑎, 𝑏⟩ ∈ V → (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑎, 𝑏⟩ ∈ 𝐵))
2018, 19ax-mp 5 . . . . 5 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑎, 𝑏⟩ ∈ 𝐵)
2115, 16opelcnv 5726 . . . . . 6 (⟨𝑎, 𝑏⟩ ∈ 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝐵)
2221ralbii 3097 . . . . 5 (∀𝑥𝐴𝑎, 𝑏⟩ ∈ 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵)
2320, 22bitri 278 . . . 4 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵)
2414, 17, 233bitr4i 306 . . 3 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵)
2524eqrelriv 5635 . 2 ((Rel 𝑥𝐴 𝐵 ∧ Rel 𝑥𝐴 𝐵) → 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
261, 11, 25sylancr 590 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ⊆ wss 3860  ∅c0 4227  ⟨cop 4531  ∩ ciin 4887   × cxp 5525  ◡ccnv 5526  Rel wrel 5532 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-iin 4889  df-br 5036  df-opab 5098  df-xp 5533  df-rel 5534  df-cnv 5535 This theorem is referenced by: (None)
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