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Theorem cnviin 6308
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 6125 . 2 Rel 𝑥𝐴 𝐵
2 relcnv 6125 . . . . . . 7 Rel 𝐵
3 df-rel 5696 . . . . . . 7 (Rel 𝐵𝐵 ⊆ (V × V))
42, 3mpbi 230 . . . . . 6 𝐵 ⊆ (V × V)
54rgenw 3063 . . . . 5 𝑥𝐴 𝐵 ⊆ (V × V)
6 r19.2z 4501 . . . . 5 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ⊆ (V × V)) → ∃𝑥𝐴 𝐵 ⊆ (V × V))
75, 6mpan2 691 . . . 4 (𝐴 ≠ ∅ → ∃𝑥𝐴 𝐵 ⊆ (V × V))
8 iinss 5061 . . . 4 (∃𝑥𝐴 𝐵 ⊆ (V × V) → 𝑥𝐴 𝐵 ⊆ (V × V))
97, 8syl 17 . . 3 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 ⊆ (V × V))
10 df-rel 5696 . . 3 (Rel 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ (V × V))
119, 10sylibr 234 . 2 (𝐴 ≠ ∅ → Rel 𝑥𝐴 𝐵)
12 opex 5475 . . . . 5 𝑏, 𝑎⟩ ∈ V
13 eliin 5001 . . . . 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 3482 . . . . 5 𝑎 ∈ V
16 vex 3482 . . . . 5 𝑏 ∈ V
1715, 16opelcnv 5895 . . . 4 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑥𝐴 𝐵)
18 opex 5475 . . . . . 6 𝑎, 𝑏⟩ ∈ V
19 eliin 5001 . . . . . 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 5895 . . . . . 6 (⟨𝑎, 𝑏⟩ ∈ 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝐵)
2221ralbii 3091 . . . . 5 (∀𝑥𝐴𝑎, 𝑏⟩ ∈ 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵)
2320, 22bitri 275 . . . 4 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵)
2414, 17, 233bitr4i 303 . . 3 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵)
2524eqrelriv 5802 . 2 ((Rel 𝑥𝐴 𝐵 ∧ Rel 𝑥𝐴 𝐵) → 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
261, 11, 25sylancr 587 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  wss 3963  c0 4339  cop 4637   ciin 4997   × cxp 5687  ccnv 5688  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-iin 4999  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697
This theorem is referenced by: (None)
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