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Theorem dfco2 6066
 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 6065 . 2 Rel (𝐴𝐵)
2 reliun 5654 . . 3 (Rel 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∀𝑥 ∈ V Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
3 relxp 5538 . . . 4 Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
43a1i 11 . . 3 (𝑥 ∈ V → Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
52, 4mprgbir 3121 . 2 Rel 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
6 opelco2g 5703 . . . 4 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴)))
76el2v 3448 . . 3 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
8 eliun 4886 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
9 rexv 3467 . . . 4 (∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
10 opelxp 5556 . . . . . 6 (⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑦 ∈ (𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})))
11 vex 3444 . . . . . . . . 9 𝑥 ∈ V
12 vex 3444 . . . . . . . . 9 𝑦 ∈ V
1311, 12elimasn 5922 . . . . . . . 8 (𝑦 ∈ (𝐵 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
1411, 12opelcnv 5717 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1513, 14bitri 278 . . . . . . 7 (𝑦 ∈ (𝐵 “ {𝑥}) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
16 vex 3444 . . . . . . . 8 𝑧 ∈ V
1711, 16elimasn 5922 . . . . . . 7 (𝑧 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
1815, 17anbi12i 629 . . . . . 6 ((𝑦 ∈ (𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
1910, 18bitri 278 . . . . 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 301 . . 3 (∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
227, 21bitri 278 . 2 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
231, 5, 22eqrelriiv 5628 1 (𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441  {csn 4525  ⟨cop 4531  ∪ ciun 4882   × cxp 5518  ◡ccnv 5519   “ cima 5523   ∘ ccom 5524  Rel wrel 5525 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-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-iun 4884  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533 This theorem is referenced by:  dfco2a  6067
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