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Theorem elxp5 4859
Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4858 when the double intersection does not create class existence problems (caused by int0 3670). (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 elex 2619 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2 elex 2619 . . . 4 ( 𝐴𝐵 𝐴 ∈ V)
3 elex 2619 . . . 4 ( ran {𝐴} ∈ 𝐶 ran {𝐴} ∈ V)
42, 3anim12i 331 . . 3 (( 𝐴𝐵 ran {𝐴} ∈ 𝐶) → ( 𝐴 ∈ V ∧ ran {𝐴} ∈ V))
5 opexg 4011 . . . . 5 (( 𝐴 ∈ V ∧ ran {𝐴} ∈ V) → ⟨ 𝐴, ran {𝐴}⟩ ∈ V)
65adantl 271 . . . 4 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴 ∈ V ∧ ran {𝐴} ∈ V)) → ⟨ 𝐴, ran {𝐴}⟩ ∈ V)
7 eleq1 2145 . . . . 5 (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ → (𝐴 ∈ V ↔ ⟨ 𝐴, ran {𝐴}⟩ ∈ V))
87adantr 270 . . . 4 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴 ∈ V ∧ ran {𝐴} ∈ V)) → (𝐴 ∈ V ↔ ⟨ 𝐴, ran {𝐴}⟩ ∈ V))
96, 8mpbird 165 . . 3 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴 ∈ V ∧ ran {𝐴} ∈ V)) → 𝐴 ∈ V)
104, 9sylan2 280 . 2 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)) → 𝐴 ∈ V)
11 elxp 4408 . . . 4 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
12 sneq 3427 . . . . . . . . . . . . . 14 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
1312rneqd 4611 . . . . . . . . . . . . 13 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
1413unieqd 3632 . . . . . . . . . . . 12 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
15 vex 2613 . . . . . . . . . . . . 13 𝑥 ∈ V
16 vex 2613 . . . . . . . . . . . . 13 𝑦 ∈ V
1715, 16op2nda 4855 . . . . . . . . . . . 12 ran {⟨𝑥, 𝑦⟩} = 𝑦
1814, 17syl6req 2132 . . . . . . . . . . 11 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = ran {𝐴})
1918pm4.71ri 384 . . . . . . . . . 10 (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
2019anbi1i 446 . . . . . . . . 9 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ((𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥𝐵𝑦𝐶)))
21 anass 393 . . . . . . . . 9 (((𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
2220, 21bitri 182 . . . . . . . 8 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
2322exbii 1537 . . . . . . 7 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑦(𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
24 snexg 3976 . . . . . . . . . 10 (𝐴 ∈ V → {𝐴} ∈ V)
25 rnexg 4645 . . . . . . . . . 10 ({𝐴} ∈ V → ran {𝐴} ∈ V)
2624, 25syl 14 . . . . . . . . 9 (𝐴 ∈ V → ran {𝐴} ∈ V)
27 uniexg 4221 . . . . . . . . 9 (ran {𝐴} ∈ V → ran {𝐴} ∈ V)
2826, 27syl 14 . . . . . . . 8 (𝐴 ∈ V → ran {𝐴} ∈ V)
29 opeq2 3591 . . . . . . . . . . 11 (𝑦 = ran {𝐴} → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ran {𝐴}⟩)
3029eqeq2d 2094 . . . . . . . . . 10 (𝑦 = ran {𝐴} → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, ran {𝐴}⟩))
31 eleq1 2145 . . . . . . . . . . 11 (𝑦 = ran {𝐴} → (𝑦𝐶 ran {𝐴} ∈ 𝐶))
3231anbi2d 452 . . . . . . . . . 10 (𝑦 = ran {𝐴} → ((𝑥𝐵𝑦𝐶) ↔ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))
3330, 32anbi12d 457 . . . . . . . . 9 (𝑦 = ran {𝐴} → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
3433ceqsexgv 2732 . . . . . . . 8 ( ran {𝐴} ∈ V → (∃𝑦(𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
3528, 34syl 14 . . . . . . 7 (𝐴 ∈ V → (∃𝑦(𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
3623, 35syl5bb 190 . . . . . 6 (𝐴 ∈ V → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
37 inteq 3659 . . . . . . . . . . . 12 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝐴 = 𝑥, ran {𝐴}⟩)
3837inteqd 3661 . . . . . . . . . . 11 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝐴 = 𝑥, ran {𝐴}⟩)
3938adantl 271 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩) → 𝐴 = 𝑥, ran {𝐴}⟩)
40 op1stbg 4256 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ ran {𝐴} ∈ V) → 𝑥, ran {𝐴}⟩ = 𝑥)
4115, 28, 40sylancr 405 . . . . . . . . . . 11 (𝐴 ∈ V → 𝑥, ran {𝐴}⟩ = 𝑥)
4241adantr 270 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩) → 𝑥, ran {𝐴}⟩ = 𝑥)
4339, 42eqtr2d 2116 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩) → 𝑥 = 𝐴)
4443ex 113 . . . . . . . 8 (𝐴 ∈ V → (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝑥 = 𝐴))
4544pm4.71rd 386 . . . . . . 7 (𝐴 ∈ V → (𝐴 = ⟨𝑥, ran {𝐴}⟩ ↔ (𝑥 = 𝐴𝐴 = ⟨𝑥, ran {𝐴}⟩)))
4645anbi1d 453 . . . . . 6 (𝐴 ∈ V → ((𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ ((𝑥 = 𝐴𝐴 = ⟨𝑥, ran {𝐴}⟩) ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
47 anass 393 . . . . . . 7 (((𝑥 = 𝐴𝐴 = ⟨𝑥, ran {𝐴}⟩) ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ (𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
4847a1i 9 . . . . . 6 (𝐴 ∈ V → (((𝑥 = 𝐴𝐴 = ⟨𝑥, ran {𝐴}⟩) ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ (𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))))
4936, 46, 483bitrd 212 . . . . 5 (𝐴 ∈ V → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))))
5049exbidv 1748 . . . 4 (𝐴 ∈ V → (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))))
5111, 50syl5bb 190 . . 3 (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))))
52 eqvisset 2618 . . . . . 6 (𝑥 = 𝐴 𝐴 ∈ V)
5352adantr 270 . . . . 5 ((𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) → 𝐴 ∈ V)
5453exlimiv 1530 . . . 4 (∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) → 𝐴 ∈ V)
552ad2antrl 474 . . . 4 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)) → 𝐴 ∈ V)
56 opeq1 3590 . . . . . . 7 (𝑥 = 𝐴 → ⟨𝑥, ran {𝐴}⟩ = ⟨ 𝐴, ran {𝐴}⟩)
5756eqeq2d 2094 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = ⟨𝑥, ran {𝐴}⟩ ↔ 𝐴 = ⟨ 𝐴, ran {𝐴}⟩))
58 eleq1 2145 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐵 𝐴𝐵))
5958anbi1d 453 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝐵 ran {𝐴} ∈ 𝐶) ↔ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
6057, 59anbi12d 457 . . . . 5 (𝑥 = 𝐴 → ((𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶))))
6160ceqsexgv 2732 . . . 4 ( 𝐴 ∈ V → (∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶))))
6254, 55, 61pm5.21nii 653 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
6351, 62syl6bb 194 . 2 (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶))))
641, 10, 63pm5.21nii 653 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  Vcvv 2610  {csn 3416  cop 3419   cuni 3621   cint 3656   × cxp 4389  ran crn 4392
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402
This theorem is referenced by: (None)
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