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Theorem elxp5 5113
Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5112 when the double intersection does not create class existence problems (caused by int0 3856). (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 2748 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2 elex 2748 . . . 4 ( 𝐴𝐵 𝐴 ∈ V)
3 elex 2748 . . . 4 ( ran {𝐴} ∈ 𝐶 ran {𝐴} ∈ V)
42, 3anim12i 338 . . 3 (( 𝐴𝐵 ran {𝐴} ∈ 𝐶) → ( 𝐴 ∈ V ∧ ran {𝐴} ∈ V))
5 opexg 4225 . . . . 5 (( 𝐴 ∈ V ∧ ran {𝐴} ∈ V) → ⟨ 𝐴, ran {𝐴}⟩ ∈ V)
65adantl 277 . . . 4 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴 ∈ V ∧ ran {𝐴} ∈ V)) → ⟨ 𝐴, ran {𝐴}⟩ ∈ V)
7 eleq1 2240 . . . . 5 (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ → (𝐴 ∈ V ↔ ⟨ 𝐴, ran {𝐴}⟩ ∈ V))
87adantr 276 . . . 4 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴 ∈ V ∧ ran {𝐴} ∈ V)) → (𝐴 ∈ V ↔ ⟨ 𝐴, ran {𝐴}⟩ ∈ V))
96, 8mpbird 167 . . 3 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴 ∈ V ∧ ran {𝐴} ∈ V)) → 𝐴 ∈ V)
104, 9sylan2 286 . 2 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)) → 𝐴 ∈ V)
11 elxp 4640 . . . 4 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
12 sneq 3602 . . . . . . . . . . . . . 14 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
1312rneqd 4852 . . . . . . . . . . . . 13 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
1413unieqd 3818 . . . . . . . . . . . 12 (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
15 vex 2740 . . . . . . . . . . . . 13 𝑥 ∈ V
16 vex 2740 . . . . . . . . . . . . 13 𝑦 ∈ V
1715, 16op2nda 5109 . . . . . . . . . . . 12 ran {⟨𝑥, 𝑦⟩} = 𝑦
1814, 17eqtr2di 2227 . . . . . . . . . . 11 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = ran {𝐴})
1918pm4.71ri 392 . . . . . . . . . 10 (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
2019anbi1i 458 . . . . . . . . 9 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ((𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥𝐵𝑦𝐶)))
21 anass 401 . . . . . . . . 9 (((𝑦 = ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
2220, 21bitri 184 . . . . . . . 8 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
2322exbii 1605 . . . . . . 7 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑦(𝑦 = ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
24 snexg 4181 . . . . . . . . . 10 (𝐴 ∈ V → {𝐴} ∈ V)
25 rnexg 4888 . . . . . . . . . 10 ({𝐴} ∈ V → ran {𝐴} ∈ V)
2624, 25syl 14 . . . . . . . . 9 (𝐴 ∈ V → ran {𝐴} ∈ V)
27 uniexg 4436 . . . . . . . . 9 (ran {𝐴} ∈ V → ran {𝐴} ∈ V)
2826, 27syl 14 . . . . . . . 8 (𝐴 ∈ V → ran {𝐴} ∈ V)
29 opeq2 3777 . . . . . . . . . . 11 (𝑦 = ran {𝐴} → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ran {𝐴}⟩)
3029eqeq2d 2189 . . . . . . . . . 10 (𝑦 = ran {𝐴} → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, ran {𝐴}⟩))
31 eleq1 2240 . . . . . . . . . . 11 (𝑦 = ran {𝐴} → (𝑦𝐶 ran {𝐴} ∈ 𝐶))
3231anbi2d 464 . . . . . . . . . 10 (𝑦 = ran {𝐴} → ((𝑥𝐵𝑦𝐶) ↔ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))
3330, 32anbi12d 473 . . . . . . . . 9 (𝑦 = ran {𝐴} → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
3433ceqsexgv 2866 . . . . . . . 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, 35bitrid 192 . . . . . 6 (𝐴 ∈ V → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
37 inteq 3845 . . . . . . . . . . . 12 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝐴 = 𝑥, ran {𝐴}⟩)
3837inteqd 3847 . . . . . . . . . . 11 (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝐴 = 𝑥, ran {𝐴}⟩)
3938adantl 277 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩) → 𝐴 = 𝑥, ran {𝐴}⟩)
40 op1stbg 4476 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ ran {𝐴} ∈ V) → 𝑥, ran {𝐴}⟩ = 𝑥)
4115, 28, 40sylancr 414 . . . . . . . . . . 11 (𝐴 ∈ V → 𝑥, ran {𝐴}⟩ = 𝑥)
4241adantr 276 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩) → 𝑥, ran {𝐴}⟩ = 𝑥)
4339, 42eqtr2d 2211 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ran {𝐴}⟩) → 𝑥 = 𝐴)
4443ex 115 . . . . . . . 8 (𝐴 ∈ V → (𝐴 = ⟨𝑥, ran {𝐴}⟩ → 𝑥 = 𝐴))
4544pm4.71rd 394 . . . . . . 7 (𝐴 ∈ V → (𝐴 = ⟨𝑥, ran {𝐴}⟩ ↔ (𝑥 = 𝐴𝐴 = ⟨𝑥, ran {𝐴}⟩)))
4645anbi1d 465 . . . . . 6 (𝐴 ∈ V → ((𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ ((𝑥 = 𝐴𝐴 = ⟨𝑥, ran {𝐴}⟩) ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))))
47 anass 401 . . . . . . 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 214 . . . . 5 (𝐴 ∈ V → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))))
5049exbidv 1825 . . . 4 (𝐴 ∈ V → (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))))
5111, 50bitrid 192 . . 3 (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)))))
52 eqvisset 2747 . . . . . 6 (𝑥 = 𝐴 𝐴 ∈ V)
5352adantr 276 . . . . 5 ((𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) → 𝐴 ∈ V)
5453exlimiv 1598 . . . 4 (∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) → 𝐴 ∈ V)
552ad2antrl 490 . . . 4 ((𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)) → 𝐴 ∈ V)
56 opeq1 3776 . . . . . . 7 (𝑥 = 𝐴 → ⟨𝑥, ran {𝐴}⟩ = ⟨ 𝐴, ran {𝐴}⟩)
5756eqeq2d 2189 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = ⟨𝑥, ran {𝐴}⟩ ↔ 𝐴 = ⟨ 𝐴, ran {𝐴}⟩))
58 eleq1 2240 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐵 𝐴𝐵))
5958anbi1d 465 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝐵 ran {𝐴} ∈ 𝐶) ↔ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
6057, 59anbi12d 473 . . . . 5 (𝑥 = 𝐴 → ((𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶)) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶))))
6160ceqsexgv 2866 . . . 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 704 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝐴 = ⟨𝑥, ran {𝐴}⟩ ∧ (𝑥𝐵 ran {𝐴} ∈ 𝐶))) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
6351, 62bitrdi 196 . 2 (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶))))
641, 10, 63pm5.21nii 704 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  Vcvv 2737  {csn 3591  cop 3594   cuni 3807   cint 3842   × cxp 4621  ran crn 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-xp 4629  df-rel 4630  df-cnv 4631  df-dm 4633  df-rn 4634
This theorem is referenced by: (None)
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