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Theorem dfco2 4917
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 4916 . 2 Rel (𝐴𝐵)
2 reliun 4546 . . 3 (Rel 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∀𝑥 ∈ V Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
3 relxp 4535 . . . 4 Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
43a1i 9 . . 3 (𝑥 ∈ V → Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
52, 4mprgbir 2433 . 2 Rel 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
6 vex 2622 . . . 4 𝑦 ∈ V
7 vex 2622 . . . 4 𝑧 ∈ V
8 opelco2g 4592 . . . 4 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴)))
96, 7, 8mp2an 417 . . 3 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
10 eliun 3729 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
11 rexv 2637 . . . 4 (∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
12 opelxp 4457 . . . . . 6 (⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑦 ∈ (𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})))
13 vex 2622 . . . . . . . . 9 𝑥 ∈ V
1413, 6elimasn 4786 . . . . . . . 8 (𝑦 ∈ (𝐵 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
1513, 6opelcnv 4606 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1614, 15bitri 182 . . . . . . 7 (𝑦 ∈ (𝐵 “ {𝑥}) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1713, 7elimasn 4786 . . . . . . 7 (𝑧 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
1816, 17anbi12i 448 . . . . . 6 ((𝑦 ∈ (𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
1912, 18bitri 182 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
2019exbii 1541 . . . 4 (∃𝑥𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
2110, 11, 203bitrri 205 . . 3 (∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
229, 21bitri 182 . 2 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
231, 5, 22eqrelriiv 4520 1 (𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  wrex 2360  Vcvv 2619  {csn 3441  cop 3444   ciun 3725   × cxp 4426  ccnv 4427  cima 4431  ccom 4432  Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-iun 3727  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441
This theorem is referenced by:  dfco2a  4918
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