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Theorem dfco2 6244
Description: Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
Assertion
Ref Expression
dfco2 (𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfco2
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 6107 . 2 Rel (𝐴𝐵)
2 reliun 5816 . . 3 (Rel 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∀𝑥 ∈ V Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
3 relxp 5694 . . . 4 Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
43a1i 11 . . 3 (𝑥 ∈ V → Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
52, 4mprgbir 3067 . 2 Rel 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
6 opelco2g 5867 . . . 4 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴)))
76el2v 3481 . . 3 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
8 eliun 5001 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
9 rexv 3499 . . . 4 (∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
10 opelxp 5712 . . . . . 6 (⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑦 ∈ (𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})))
11 vex 3477 . . . . . . . . 9 𝑥 ∈ V
12 vex 3477 . . . . . . . . 9 𝑦 ∈ V
1311, 12elimasn 6088 . . . . . . . 8 (𝑦 ∈ (𝐵 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
1411, 12opelcnv 5881 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1513, 14bitri 275 . . . . . . 7 (𝑦 ∈ (𝐵 “ {𝑥}) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
16 vex 3477 . . . . . . . 8 𝑧 ∈ V
1711, 16elimasn 6088 . . . . . . 7 (𝑧 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
1815, 17anbi12i 626 . . . . . 6 ((𝑦 ∈ (𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
1910, 18bitri 275 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
2019exbii 1849 . . . 4 (∃𝑥𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
218, 9, 203bitrri 298 . . 3 (∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
227, 21bitri 275 . 2 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
231, 5, 22eqrelriiv 5790 1 (𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1540  wex 1780  wcel 2105  wrex 3069  Vcvv 3473  {csn 4628  cop 4634   ciun 4997   × cxp 5674  ccnv 5675  cima 5679  ccom 5680  Rel wrel 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-iun 4999  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  dfco2a  6245
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