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Theorem elxp4 7057
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp5 7058, elxp6 7145, and elxp7 7146. (Contributed by NM, 17-Feb-2004.)
Assertion
Ref Expression
elxp4 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))

Proof of Theorem elxp4
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 5091 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
2 sneq 4158 . . . . . . . . . . . 12 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
32rneqd 5313 . . . . . . . . . . 11 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
43unieqd 4412 . . . . . . . . . 10 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
5 vex 3189 . . . . . . . . . . 11 𝑥 ∈ V
6 vex 3189 . . . . . . . . . . 11 𝑦 ∈ V
75, 6op2nda 5579 . . . . . . . . . 10 ran {⟨𝑥, 𝑦⟩} = 𝑦
84, 7syl6req 2672 . . . . . . . . 9 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = ran {𝐴})
98pm4.71ri 664 . . . . . . . 8 (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
109anbi1i 730 . . . . . . 7 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ((𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥𝐵𝑦𝐶)))
11 anass 680 . . . . . . 7 (((𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
1210, 11bitri 264 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
1312exbii 1771 . . . . 5 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑦(𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
14 snex 4869 . . . . . . . 8 {𝐴} ∈ V
1514rnex 7047 . . . . . . 7 ran {𝐴} ∈ V
1615uniex 6906 . . . . . 6 ran {𝐴} ∈ V
17 opeq2 4371 . . . . . . . 8 (𝑦 = ran {𝐴} → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ran {𝐴}⟩)
1817eqeq2d 2631 . . . . . . 7 (𝑦 = ran {𝐴} → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, ran {𝐴}⟩))
19 eleq1 2686 . . . . . . . 8 (𝑦 = ran {𝐴} → (𝑦𝐶 ran {𝐴} ∈ 𝐶))
2019anbi2d 739 . . . . . . 7 (𝑦 = ran {𝐴} → ((𝑥𝐵𝑦𝐶) ↔ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))
2118, 20anbi12d 746 . . . . . 6 (𝑦 = ran {𝐴} → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
2216, 21ceqsexv 3228 . . . . 5 (∃𝑦(𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))
2313, 22bitri 264 . . . 4 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))
24 sneq 4158 . . . . . . . . 9 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → {𝐴} = {⟨𝑥, ran {𝐴}⟩})
2524dmeqd 5286 . . . . . . . 8 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → dom {𝐴} = dom {⟨𝑥, ran {𝐴}⟩})
2625unieqd 4412 . . . . . . 7 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → dom {𝐴} = dom {⟨𝑥, ran {𝐴}⟩})
275, 16op1sta 5576 . . . . . . 7 dom {⟨𝑥, ran {𝐴}⟩} = 𝑥
2826, 27syl6req 2672 . . . . . 6 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝑥 = dom {𝐴})
2928pm4.71ri 664 . . . . 5 (𝐴 = ⟨𝑥, ran {𝐴}⟩ ↔ (𝑥 = dom {𝐴} ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩))
3029anbi1i 730 . . . 4 ((𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ ((𝑥 = dom {𝐴} ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩) ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))
31 anass 680 . . . 4 (((𝑥 = dom {𝐴} ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩) ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ (𝑥 = dom {𝐴} ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
3223, 30, 313bitri 286 . . 3 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑥 = dom {𝐴} ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
3332exbii 1771 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑥(𝑥 = dom {𝐴} ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
3414dmex 7046 . . . 4 dom {𝐴} ∈ V
3534uniex 6906 . . 3 dom {𝐴} ∈ V
36 opeq1 4370 . . . . 5 (𝑥 = dom {𝐴} → ⟨𝑥, ran {𝐴}⟩ = ⟨ dom {𝐴}, ran {𝐴}⟩)
3736eqeq2d 2631 . . . 4 (𝑥 = dom {𝐴} → (𝐴 = ⟨𝑥, ran {𝐴}⟩ ↔ 𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩))
38 eleq1 2686 . . . . 5 (𝑥 = dom {𝐴} → (𝑥𝐵 dom {𝐴} ∈ 𝐵))
3938anbi1d 740 . . . 4 (𝑥 = dom {𝐴} → ((𝑥𝐵 ran {𝐴} ∈ 𝐶) ↔ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
4037, 39anbi12d 746 . . 3 (𝑥 = dom {𝐴} → ((𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶))))
4135, 40ceqsexv 3228 . 2 (∃𝑥(𝑥 = dom {𝐴} ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
421, 33, 413bitri 286 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  {csn 4148  cop 4154   cuni 4402   × cxp 5072  dom cdm 5074  ran crn 5075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085
This theorem is referenced by:  elxp6  7145  xpdom2  7999
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