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Theorem elxp4 4751
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4752. (Contributed by NM, 17-Feb-2004.)
Assertion
Ref Expression
elxp4 (A (B × 𝐶) ↔ (A = ⟨ dom {A}, ran {A}⟩ ( dom {A} B ran {A} 𝐶)))

Proof of Theorem elxp4
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2560 . 2 (A (B × 𝐶) → A V)
2 elex 2560 . . . 4 ( dom {A} B dom {A} V)
3 elex 2560 . . . 4 ( ran {A} 𝐶 ran {A} V)
42, 3anim12i 321 . . 3 (( dom {A} B ran {A} 𝐶) → ( dom {A} V ran {A} V))
5 opexgOLD 3956 . . . . 5 (( dom {A} V ran {A} V) → ⟨ dom {A}, ran {A}⟩ V)
65adantl 262 . . . 4 ((A = ⟨ dom {A}, ran {A}⟩ ( dom {A} V ran {A} V)) → ⟨ dom {A}, ran {A}⟩ V)
7 eleq1 2097 . . . . 5 (A = ⟨ dom {A}, ran {A}⟩ → (A V ↔ ⟨ dom {A}, ran {A}⟩ V))
87adantr 261 . . . 4 ((A = ⟨ dom {A}, ran {A}⟩ ( dom {A} V ran {A} V)) → (A V ↔ ⟨ dom {A}, ran {A}⟩ V))
96, 8mpbird 156 . . 3 ((A = ⟨ dom {A}, ran {A}⟩ ( dom {A} V ran {A} V)) → A V)
104, 9sylan2 270 . 2 ((A = ⟨ dom {A}, ran {A}⟩ ( dom {A} B ran {A} 𝐶)) → A V)
11 elxp 4305 . . . 4 (A (B × 𝐶) ↔ xy(A = ⟨x, y (x B y 𝐶)))
1211a1i 9 . . 3 (A V → (A (B × 𝐶) ↔ xy(A = ⟨x, y (x B y 𝐶))))
13 sneq 3378 . . . . . . . . . . . . 13 (A = ⟨x, y⟩ → {A} = {⟨x, y⟩})
1413rneqd 4506 . . . . . . . . . . . 12 (A = ⟨x, y⟩ → ran {A} = ran {⟨x, y⟩})
1514unieqd 3582 . . . . . . . . . . 11 (A = ⟨x, y⟩ → ran {A} = ran {⟨x, y⟩})
16 vex 2554 . . . . . . . . . . . 12 x V
17 vex 2554 . . . . . . . . . . . 12 y V
1816, 17op2nda 4748 . . . . . . . . . . 11 ran {⟨x, y⟩} = y
1915, 18syl6req 2086 . . . . . . . . . 10 (A = ⟨x, y⟩ → y = ran {A})
2019pm4.71ri 372 . . . . . . . . 9 (A = ⟨x, y⟩ ↔ (y = ran {A} A = ⟨x, y⟩))
2120anbi1i 431 . . . . . . . 8 ((A = ⟨x, y (x B y 𝐶)) ↔ ((y = ran {A} A = ⟨x, y⟩) (x B y 𝐶)))
22 anass 381 . . . . . . . 8 (((y = ran {A} A = ⟨x, y⟩) (x B y 𝐶)) ↔ (y = ran {A} (A = ⟨x, y (x B y 𝐶))))
2321, 22bitri 173 . . . . . . 7 ((A = ⟨x, y (x B y 𝐶)) ↔ (y = ran {A} (A = ⟨x, y (x B y 𝐶))))
2423exbii 1493 . . . . . 6 (y(A = ⟨x, y (x B y 𝐶)) ↔ y(y = ran {A} (A = ⟨x, y (x B y 𝐶))))
25 snexgOLD 3926 . . . . . . . . 9 (A V → {A} V)
26 rnexg 4540 . . . . . . . . 9 ({A} V → ran {A} V)
2725, 26syl 14 . . . . . . . 8 (A V → ran {A} V)
28 uniexg 4141 . . . . . . . 8 (ran {A} V → ran {A} V)
2927, 28syl 14 . . . . . . 7 (A V → ran {A} V)
30 opeq2 3541 . . . . . . . . . 10 (y = ran {A} → ⟨x, y⟩ = ⟨x, ran {A}⟩)
3130eqeq2d 2048 . . . . . . . . 9 (y = ran {A} → (A = ⟨x, y⟩ ↔ A = ⟨x, ran {A}⟩))
32 eleq1 2097 . . . . . . . . . 10 (y = ran {A} → (y 𝐶 ran {A} 𝐶))
3332anbi2d 437 . . . . . . . . 9 (y = ran {A} → ((x B y 𝐶) ↔ (x B ran {A} 𝐶)))
3431, 33anbi12d 442 . . . . . . . 8 (y = ran {A} → ((A = ⟨x, y (x B y 𝐶)) ↔ (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))))
3534ceqsexgv 2667 . . . . . . 7 ( ran {A} V → (y(y = ran {A} (A = ⟨x, y (x B y 𝐶))) ↔ (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))))
3629, 35syl 14 . . . . . 6 (A V → (y(y = ran {A} (A = ⟨x, y (x B y 𝐶))) ↔ (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))))
3724, 36syl5bb 181 . . . . 5 (A V → (y(A = ⟨x, y (x B y 𝐶)) ↔ (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))))
38 sneq 3378 . . . . . . . . . . . 12 (A = ⟨x, ran {A}⟩ → {A} = {⟨x, ran {A}⟩})
3938dmeqd 4480 . . . . . . . . . . 11 (A = ⟨x, ran {A}⟩ → dom {A} = dom {⟨x, ran {A}⟩})
4039unieqd 3582 . . . . . . . . . 10 (A = ⟨x, ran {A}⟩ → dom {A} = dom {⟨x, ran {A}⟩})
4140adantl 262 . . . . . . . . 9 ((A V A = ⟨x, ran {A}⟩) → dom {A} = dom {⟨x, ran {A}⟩})
42 dmsnopg 4735 . . . . . . . . . . . . 13 ( ran {A} V → dom {⟨x, ran {A}⟩} = {x})
4329, 42syl 14 . . . . . . . . . . . 12 (A V → dom {⟨x, ran {A}⟩} = {x})
4443unieqd 3582 . . . . . . . . . . 11 (A V → dom {⟨x, ran {A}⟩} = {x})
4516unisn 3587 . . . . . . . . . . 11 {x} = x
4644, 45syl6eq 2085 . . . . . . . . . 10 (A V → dom {⟨x, ran {A}⟩} = x)
4746adantr 261 . . . . . . . . 9 ((A V A = ⟨x, ran {A}⟩) → dom {⟨x, ran {A}⟩} = x)
4841, 47eqtr2d 2070 . . . . . . . 8 ((A V A = ⟨x, ran {A}⟩) → x = dom {A})
4948ex 108 . . . . . . 7 (A V → (A = ⟨x, ran {A}⟩ → x = dom {A}))
5049pm4.71rd 374 . . . . . 6 (A V → (A = ⟨x, ran {A}⟩ ↔ (x = dom {A} A = ⟨x, ran {A}⟩)))
5150anbi1d 438 . . . . 5 (A V → ((A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶)) ↔ ((x = dom {A} A = ⟨x, ran {A}⟩) (x B ran {A} 𝐶))))
52 anass 381 . . . . . 6 (((x = dom {A} A = ⟨x, ran {A}⟩) (x B ran {A} 𝐶)) ↔ (x = dom {A} (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))))
5352a1i 9 . . . . 5 (A V → (((x = dom {A} A = ⟨x, ran {A}⟩) (x B ran {A} 𝐶)) ↔ (x = dom {A} (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶)))))
5437, 51, 533bitrd 203 . . . 4 (A V → (y(A = ⟨x, y (x B y 𝐶)) ↔ (x = dom {A} (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶)))))
5554exbidv 1703 . . 3 (A V → (xy(A = ⟨x, y (x B y 𝐶)) ↔ x(x = dom {A} (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶)))))
56 dmexg 4539 . . . . . 6 ({A} V → dom {A} V)
5725, 56syl 14 . . . . 5 (A V → dom {A} V)
58 uniexg 4141 . . . . 5 (dom {A} V → dom {A} V)
5957, 58syl 14 . . . 4 (A V → dom {A} V)
60 opeq1 3540 . . . . . . 7 (x = dom {A} → ⟨x, ran {A}⟩ = ⟨ dom {A}, ran {A}⟩)
6160eqeq2d 2048 . . . . . 6 (x = dom {A} → (A = ⟨x, ran {A}⟩ ↔ A = ⟨ dom {A}, ran {A}⟩))
62 eleq1 2097 . . . . . . 7 (x = dom {A} → (x B dom {A} B))
6362anbi1d 438 . . . . . 6 (x = dom {A} → ((x B ran {A} 𝐶) ↔ ( dom {A} B ran {A} 𝐶)))
6461, 63anbi12d 442 . . . . 5 (x = dom {A} → ((A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶)) ↔ (A = ⟨ dom {A}, ran {A}⟩ ( dom {A} B ran {A} 𝐶))))
6564ceqsexgv 2667 . . . 4 ( dom {A} V → (x(x = dom {A} (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))) ↔ (A = ⟨ dom {A}, ran {A}⟩ ( dom {A} B ran {A} 𝐶))))
6659, 65syl 14 . . 3 (A V → (x(x = dom {A} (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))) ↔ (A = ⟨ dom {A}, ran {A}⟩ ( dom {A} B ran {A} 𝐶))))
6712, 55, 663bitrd 203 . 2 (A V → (A (B × 𝐶) ↔ (A = ⟨ dom {A}, ran {A}⟩ ( dom {A} B ran {A} 𝐶))))
681, 10, 67pm5.21nii 619 1 (A (B × 𝐶) ↔ (A = ⟨ dom {A}, ran {A}⟩ ( dom {A} B ran {A} 𝐶)))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  {csn 3367  cop 3370   cuni 3571   × cxp 4286  dom cdm 4288  ran crn 4289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  elxp6  5738  xpdom2  6241
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