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Theorem dfco2 5041
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 5040 . 2 Rel (𝐴𝐵)
2 reliun 4663 . . 3 (Rel 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∀𝑥 ∈ V Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
3 relxp 4651 . . . 4 Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
43a1i 9 . . 3 (𝑥 ∈ V → Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
52, 4mprgbir 2490 . 2 Rel 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
6 vex 2689 . . . 4 𝑦 ∈ V
7 vex 2689 . . . 4 𝑧 ∈ V
8 opelco2g 4710 . . . 4 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴)))
96, 7, 8mp2an 422 . . 3 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
10 eliun 3820 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
11 rexv 2704 . . . 4 (∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
12 opelxp 4572 . . . . . 6 (⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑦 ∈ (𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})))
13 vex 2689 . . . . . . . . 9 𝑥 ∈ V
1413, 6elimasn 4909 . . . . . . . 8 (𝑦 ∈ (𝐵 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
1513, 6opelcnv 4724 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1614, 15bitri 183 . . . . . . 7 (𝑦 ∈ (𝐵 “ {𝑥}) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1713, 7elimasn 4909 . . . . . . 7 (𝑧 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
1816, 17anbi12i 455 . . . . . 6 ((𝑦 ∈ (𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
1912, 18bitri 183 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
2019exbii 1584 . . . 4 (∃𝑥𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
2110, 11, 203bitrri 206 . . 3 (∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
229, 21bitri 183 . 2 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
231, 5, 22eqrelriiv 4636 1 (𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1331  wex 1468  wcel 1480  wrex 2417  Vcvv 2686  {csn 3527  cop 3530   ciun 3816   × cxp 4540  ccnv 4541  cima 4545  ccom 4546  Rel wrel 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-iun 3818  df-br 3933  df-opab 3993  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555
This theorem is referenced by:  dfco2a  5042
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