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Theorem dfco2 5110
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 5109 . 2 Rel (𝐴𝐵)
2 reliun 4732 . . 3 (Rel 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∀𝑥 ∈ V Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
3 relxp 4720 . . . 4 Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
43a1i 9 . . 3 (𝑥 ∈ V → Rel ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
52, 4mprgbir 2528 . 2 Rel 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
6 vex 2733 . . . 4 𝑦 ∈ V
7 vex 2733 . . . 4 𝑧 ∈ V
8 opelco2g 4779 . . . 4 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴)))
96, 7, 8mp2an 424 . . 3 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
10 eliun 3877 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
11 rexv 2748 . . . 4 (∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
12 opelxp 4641 . . . . . 6 (⟨𝑦, 𝑧⟩ ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑦 ∈ (𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})))
13 vex 2733 . . . . . . . . 9 𝑥 ∈ V
1413, 6elimasn 4978 . . . . . . . 8 (𝑦 ∈ (𝐵 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
1513, 6opelcnv 4793 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1614, 15bitri 183 . . . . . . 7 (𝑦 ∈ (𝐵 “ {𝑥}) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
1713, 7elimasn 4978 . . . . . . 7 (𝑧 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
1816, 17anbi12i 457 . . . . . 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 1598 . . . 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 4705 1 (𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1348  wex 1485  wcel 2141  wrex 2449  Vcvv 2730  {csn 3583  cop 3586   ciun 3873   × cxp 4609  ccnv 4610  cima 4614  ccom 4615  Rel wrel 4616
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-iun 3875  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624
This theorem is referenced by:  dfco2a  5111
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