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Theorem elxp4 7444
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp5 7445, elxp6 7537, and elxp7 7538. (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 5431 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
2 sneq 4452 . . . . . . . . . . . 12 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
32rneqd 5652 . . . . . . . . . . 11 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
43unieqd 4723 . . . . . . . . . 10 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
5 vex 3418 . . . . . . . . . . 11 𝑥 ∈ V
6 vex 3418 . . . . . . . . . . 11 𝑦 ∈ V
75, 6op2nda 5926 . . . . . . . . . 10 ran {⟨𝑥, 𝑦⟩} = 𝑦
84, 7syl6req 2831 . . . . . . . . 9 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = ran {𝐴})
98pm4.71ri 553 . . . . . . . 8 (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
109anbi1i 614 . . . . . . 7 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ((𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥𝐵𝑦𝐶)))
11 anass 461 . . . . . . 7 (((𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
1210, 11bitri 267 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
1312exbii 1810 . . . . 5 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑦(𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
14 snex 5189 . . . . . . . 8 {𝐴} ∈ V
1514rnex 7434 . . . . . . 7 ran {𝐴} ∈ V
1615uniex 7285 . . . . . 6 ran {𝐴} ∈ V
17 opeq2 4679 . . . . . . . 8 (𝑦 = ran {𝐴} → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ran {𝐴}⟩)
1817eqeq2d 2788 . . . . . . 7 (𝑦 = ran {𝐴} → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, ran {𝐴}⟩))
19 eleq1 2853 . . . . . . . 8 (𝑦 = ran {𝐴} → (𝑦𝐶 ran {𝐴} ∈ 𝐶))
2019anbi2d 619 . . . . . . 7 (𝑦 = ran {𝐴} → ((𝑥𝐵𝑦𝐶) ↔ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))
2118, 20anbi12d 621 . . . . . 6 (𝑦 = ran {𝐴} → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
2216, 21ceqsexv 3462 . . . . 5 (∃𝑦(𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))
2313, 22bitri 267 . . . 4 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))
24 sneq 4452 . . . . . . . . 9 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → {𝐴} = {⟨𝑥, ran {𝐴}⟩})
2524dmeqd 5625 . . . . . . . 8 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → dom {𝐴} = dom {⟨𝑥, ran {𝐴}⟩})
2625unieqd 4723 . . . . . . 7 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → dom {𝐴} = dom {⟨𝑥, ran {𝐴}⟩})
275, 16op1sta 5923 . . . . . . 7 dom {⟨𝑥, ran {𝐴}⟩} = 𝑥
2826, 27syl6req 2831 . . . . . 6 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝑥 = dom {𝐴})
2928pm4.71ri 553 . . . . 5 (𝐴 = ⟨𝑥, ran {𝐴}⟩ ↔ (𝑥 = dom {𝐴} ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩))
3029anbi1i 614 . . . 4 ((𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ ((𝑥 = dom {𝐴} ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩) ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))
31 anass 461 . . . 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 289 . . 3 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑥 = dom {𝐴} ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
3332exbii 1810 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑥(𝑥 = dom {𝐴} ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
3414dmex 7433 . . . 4 dom {𝐴} ∈ V
3534uniex 7285 . . 3 dom {𝐴} ∈ V
36 opeq1 4678 . . . . 5 (𝑥 = dom {𝐴} → ⟨𝑥, ran {𝐴}⟩ = ⟨ dom {𝐴}, ran {𝐴}⟩)
3736eqeq2d 2788 . . . 4 (𝑥 = dom {𝐴} → (𝐴 = ⟨𝑥, ran {𝐴}⟩ ↔ 𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩))
38 eleq1 2853 . . . . 5 (𝑥 = dom {𝐴} → (𝑥𝐵 dom {𝐴} ∈ 𝐵))
3938anbi1d 620 . . . 4 (𝑥 = dom {𝐴} → ((𝑥𝐵 ran {𝐴} ∈ 𝐶) ↔ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
4037, 39anbi12d 621 . . 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 3462 . 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 289 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387   = wceq 1507  wex 1742  wcel 2050  {csn 4442  cop 4448   cuni 4713   × cxp 5406  dom cdm 5408  ran crn 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-xp 5414  df-rel 5415  df-cnv 5416  df-dm 5418  df-rn 5419
This theorem is referenced by:  elxp6  7537  xpdom2  8410
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