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Theorem elxp5 4836
Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4835 when the double intersection does not create class existence problems (caused by int0 3656). (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 2583 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2 elex 2583 . . . 4 ( 𝐴𝐵 𝐴 ∈ V)
3 elex 2583 . . . 4 ( ran {𝐴} ∈ 𝐶 ran {𝐴} ∈ V)
42, 3anim12i 325 . . 3 (( 𝐴𝐵 ran {𝐴} ∈ 𝐶) → ( 𝐴 ∈ V ∧ ran {𝐴} ∈ V))
5 opexgOLD 3992 . . . . 5 (( 𝐴 ∈ V ∧ ran {𝐴} ∈ V) → ⟨ 𝐴, ran {𝐴}⟩ ∈ V)
65adantl 266 . . . 4 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴 ∈ V ∧ ran {𝐴} ∈ V)) → ⟨ 𝐴, ran {𝐴}⟩ ∈ V)
7 eleq1 2116 . . . . 5 (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ → (𝐴 ∈ V ↔ ⟨ 𝐴, ran {𝐴}⟩ ∈ V))
87adantr 265 . . . 4 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴 ∈ V ∧ ran {𝐴} ∈ V)) → (𝐴 ∈ V ↔ ⟨ 𝐴, ran {𝐴}⟩ ∈ V))
96, 8mpbird 160 . . 3 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴 ∈ V ∧ ran {𝐴} ∈ V)) → 𝐴 ∈ V)
104, 9sylan2 274 . 2 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)) → 𝐴 ∈ V)
11 elxp 4389 . . . 4 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
12 sneq 3413 . . . . . . . . . . . . . 14 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
1312rneqd 4590 . . . . . . . . . . . . 13 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
1413unieqd 3618 . . . . . . . . . . . 12 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
15 vex 2577 . . . . . . . . . . . . 13 𝑥 ∈ V
16 vex 2577 . . . . . . . . . . . . 13 𝑦 ∈ V
1715, 16op2nda 4832 . . . . . . . . . . . 12 ran {⟨𝑥, 𝑦⟩} = 𝑦
1814, 17syl6req 2105 . . . . . . . . . . 11 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = ran {𝐴})
1918pm4.71ri 378 . . . . . . . . . 10 (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
2019anbi1i 439 . . . . . . . . 9 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ((𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥𝐵𝑦𝐶)))
21 anass 387 . . . . . . . . 9 (((𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
2220, 21bitri 177 . . . . . . . 8 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
2322exbii 1512 . . . . . . 7 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑦(𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
24 snexgOLD 3962 . . . . . . . . . 10 (𝐴 ∈ V → {𝐴} ∈ V)
25 rnexg 4624 . . . . . . . . . 10 ({𝐴} ∈ V → ran {𝐴} ∈ V)
2624, 25syl 14 . . . . . . . . 9 (𝐴 ∈ V → ran {𝐴} ∈ V)
27 uniexg 4202 . . . . . . . . 9 (ran {𝐴} ∈ V → ran {𝐴} ∈ V)
2826, 27syl 14 . . . . . . . 8 (𝐴 ∈ V → ran {𝐴} ∈ V)
29 opeq2 3577 . . . . . . . . . . 11 (𝑦 = ran {𝐴} → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ran {𝐴}⟩)
3029eqeq2d 2067 . . . . . . . . . 10 (𝑦 = ran {𝐴} → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, ran {𝐴}⟩))
31 eleq1 2116 . . . . . . . . . . 11 (𝑦 = ran {𝐴} → (𝑦𝐶 ran {𝐴} ∈ 𝐶))
3231anbi2d 445 . . . . . . . . . 10 (𝑦 = ran {𝐴} → ((𝑥𝐵𝑦𝐶) ↔ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))
3330, 32anbi12d 450 . . . . . . . . 9 (𝑦 = ran {𝐴} → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
3433ceqsexgv 2695 . . . . . . . 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 185 . . . . . 6 (𝐴 ∈ V → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
37 inteq 3645 . . . . . . . . . . . 12 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝐴 = 𝑥, ran {𝐴}⟩)
3837inteqd 3647 . . . . . . . . . . 11 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝐴 = 𝑥, ran {𝐴}⟩)
3938adantl 266 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩) → 𝐴 = 𝑥, ran {𝐴}⟩)
40 op1stbg 4237 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ ran {𝐴} ∈ V) → 𝑥, ran {𝐴}⟩ = 𝑥)
4115, 28, 40sylancr 399 . . . . . . . . . . 11 (𝐴 ∈ V → 𝑥, ran {𝐴}⟩ = 𝑥)
4241adantr 265 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩) → 𝑥, ran {𝐴}⟩ = 𝑥)
4339, 42eqtr2d 2089 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩) → 𝑥 = 𝐴)
4443ex 112 . . . . . . . 8 (𝐴 ∈ V → (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝑥 = 𝐴))
4544pm4.71rd 380 . . . . . . 7 (𝐴 ∈ V → (𝐴 = ⟨𝑥, ran {𝐴}⟩ ↔ (𝑥 = 𝐴𝐴 = ⟨𝑥, ran {𝐴}⟩)))
4645anbi1d 446 . . . . . 6 (𝐴 ∈ V → ((𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ ((𝑥 = 𝐴𝐴 = ⟨𝑥, ran {𝐴}⟩) ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
47 anass 387 . . . . . . 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 207 . . . . 5 (𝐴 ∈ V → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))))
5049exbidv 1722 . . . 4 (𝐴 ∈ V → (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))))
5111, 50syl5bb 185 . . 3 (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))))
52 eleq1 2116 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
5315, 52mpbii 140 . . . . . 6 (𝑥 = 𝐴 𝐴 ∈ V)
5453adantr 265 . . . . 5 ((𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) → 𝐴 ∈ V)
5554exlimiv 1505 . . . 4 (∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) → 𝐴 ∈ V)
562ad2antrl 467 . . . 4 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)) → 𝐴 ∈ V)
57 opeq1 3576 . . . . . . 7 (𝑥 = 𝐴 → ⟨𝑥, ran {𝐴}⟩ = ⟨ 𝐴, ran {𝐴}⟩)
5857eqeq2d 2067 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = ⟨𝑥, ran {𝐴}⟩ ↔ 𝐴 = ⟨ 𝐴, ran {𝐴}⟩))
59 eleq1 2116 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐵 𝐴𝐵))
6059anbi1d 446 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝐵 ran {𝐴} ∈ 𝐶) ↔ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
6158, 60anbi12d 450 . . . . 5 (𝑥 = 𝐴 → ((𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶))))
6261ceqsexgv 2695 . . . 4 ( 𝐴 ∈ V → (∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶))))
6355, 56, 62pm5.21nii 630 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
6451, 63syl6bb 189 . 2 (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶))))
651, 10, 64pm5.21nii 630 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  {csn 3402  cop 3405   cuni 3607   cint 3642   × cxp 4370  ran crn 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380  df-dm 4382  df-rn 4383
This theorem is referenced by: (None)
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