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Theorem elxp5 7073
Description: Membership in a Cartesian product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 7072 when the double intersection does not create class existence problems (caused by int0 4462). (Contributed by NM, 1-Aug-2004.)
Assertion
Ref Expression
elxp5 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))

Proof of Theorem elxp5
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 5101 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
2 sneq 4165 . . . . . . . . . . . 12 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
32rneqd 5323 . . . . . . . . . . 11 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
43unieqd 4419 . . . . . . . . . 10 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
5 vex 3193 . . . . . . . . . . 11 𝑥 ∈ V
6 vex 3193 . . . . . . . . . . 11 𝑦 ∈ V
75, 6op2nda 5589 . . . . . . . . . 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 4879 . . . . . . . 8 {𝐴} ∈ V
1514rnex 7062 . . . . . . 7 ran {𝐴} ∈ V
1615uniex 6918 . . . . . 6 ran {𝐴} ∈ V
17 opeq2 4378 . . . . . . . 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 3232 . . . . 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 inteq 4450 . . . . . . . 8 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝐴 = 𝑥, ran {𝐴}⟩)
2524inteqd 4452 . . . . . . 7 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝐴 = 𝑥, ran {𝐴}⟩)
265, 16op1stb 4911 . . . . . . 7 𝑥, ran {𝐴}⟩ = 𝑥
2725, 26syl6req 2672 . . . . . 6 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝑥 = 𝐴)
2827pm4.71ri 664 . . . . 5 (𝐴 = ⟨𝑥, ran {𝐴}⟩ ↔ (𝑥 = 𝐴𝐴 = ⟨𝑥, ran {𝐴}⟩))
2928anbi1i 730 . . . 4 ((𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ ((𝑥 = 𝐴𝐴 = ⟨𝑥, ran {𝐴}⟩) ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))
30 anass 680 . . . 4 (((𝑥 = 𝐴𝐴 = ⟨𝑥, ran {𝐴}⟩) ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ (𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
3123, 29, 303bitri 286 . . 3 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
3231exbii 1771 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
33 eqvisset 3201 . . . . 5 (𝑥 = 𝐴 𝐴 ∈ V)
3433adantr 481 . . . 4 ((𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) → 𝐴 ∈ V)
3534exlimiv 1855 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) → 𝐴 ∈ V)
36 elex 3202 . . . 4 ( 𝐴𝐵 𝐴 ∈ V)
3736ad2antrl 763 . . 3 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)) → 𝐴 ∈ V)
38 opeq1 4377 . . . . . 6 (𝑥 = 𝐴 → ⟨𝑥, ran {𝐴}⟩ = ⟨ 𝐴, ran {𝐴}⟩)
3938eqeq2d 2631 . . . . 5 (𝑥 = 𝐴 → (𝐴 = ⟨𝑥, ran {𝐴}⟩ ↔ 𝐴 = ⟨ 𝐴, ran {𝐴}⟩))
40 eleq1 2686 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐵 𝐴𝐵))
4140anbi1d 740 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐵 ran {𝐴} ∈ 𝐶) ↔ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
4239, 41anbi12d 746 . . . 4 (𝑥 = 𝐴 → ((𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶))))
4342ceqsexgv 3323 . . 3 ( 𝐴 ∈ V → (∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶))))
4435, 37, 43pm5.21nii 368 . 2 (∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
451, 32, 443bitri 286 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3190  {csn 4155  cop 4161   cuni 4409   cint 4447   × cxp 5082  ran crn 5085
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 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
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-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-dm 5094  df-rn 5095
This theorem is referenced by: (None)
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