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Theorem elxp5 4887
Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4886 when the double intersection does not create class existence problems (caused by int0 3687). (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 2624 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2 elex 2624 . . . 4 ( 𝐴𝐵 𝐴 ∈ V)
3 elex 2624 . . . 4 ( ran {𝐴} ∈ 𝐶 ran {𝐴} ∈ V)
42, 3anim12i 331 . . 3 (( 𝐴𝐵 ran {𝐴} ∈ 𝐶) → ( 𝐴 ∈ V ∧ ran {𝐴} ∈ V))
5 opexg 4031 . . . . 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 2147 . . . . 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 4430 . . . 4 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
12 sneq 3442 . . . . . . . . . . . . . 14 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
1312rneqd 4634 . . . . . . . . . . . . 13 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
1413unieqd 3649 . . . . . . . . . . . 12 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
15 vex 2618 . . . . . . . . . . . . 13 𝑥 ∈ V
16 vex 2618 . . . . . . . . . . . . 13 𝑦 ∈ V
1715, 16op2nda 4883 . . . . . . . . . . . 12 ran {⟨𝑥, 𝑦⟩} = 𝑦
1814, 17syl6req 2134 . . . . . . . . . . 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 1539 . . . . . . 7 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑦(𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
24 snexg 3995 . . . . . . . . . 10 (𝐴 ∈ V → {𝐴} ∈ V)
25 rnexg 4668 . . . . . . . . . 10 ({𝐴} ∈ V → ran {𝐴} ∈ V)
2624, 25syl 14 . . . . . . . . 9 (𝐴 ∈ V → ran {𝐴} ∈ V)
27 uniexg 4241 . . . . . . . . 9 (ran {𝐴} ∈ V → ran {𝐴} ∈ V)
2826, 27syl 14 . . . . . . . 8 (𝐴 ∈ V → ran {𝐴} ∈ V)
29 opeq2 3608 . . . . . . . . . . 11 (𝑦 = ran {𝐴} → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ran {𝐴}⟩)
3029eqeq2d 2096 . . . . . . . . . 10 (𝑦 = ran {𝐴} → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, ran {𝐴}⟩))
31 eleq1 2147 . . . . . . . . . . 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 2737 . . . . . . . 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 3676 . . . . . . . . . . . 12 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝐴 = 𝑥, ran {𝐴}⟩)
3837inteqd 3678 . . . . . . . . . . 11 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝐴 = 𝑥, ran {𝐴}⟩)
3938adantl 271 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩) → 𝐴 = 𝑥, ran {𝐴}⟩)
40 op1stbg 4276 . . . . . . . . . . . 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 2118 . . . . . . . . 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 1750 . . . 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 2623 . . . . . 6 (𝑥 = 𝐴 𝐴 ∈ V)
5352adantr 270 . . . . 5 ((𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) → 𝐴 ∈ V)
5453exlimiv 1532 . . . 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 3607 . . . . . . 7 (𝑥 = 𝐴 → ⟨𝑥, ran {𝐴}⟩ = ⟨ 𝐴, ran {𝐴}⟩)
5756eqeq2d 2096 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = ⟨𝑥, ran {𝐴}⟩ ↔ 𝐴 = ⟨ 𝐴, ran {𝐴}⟩))
58 eleq1 2147 . . . . . . 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 2737 . . . 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 1287  wex 1424  wcel 1436  Vcvv 2615  {csn 3431  cop 3434   cuni 3638   cint 3673   × cxp 4411  ran crn 4414
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-br 3823  df-opab 3877  df-xp 4419  df-rel 4420  df-cnv 4421  df-dm 4423  df-rn 4424
This theorem is referenced by: (None)
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