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Theorem elxp4 4733
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4734. (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 2542 . 2 (A (B × 𝐶) → A V)
2 elex 2542 . . . 4 ( dom {A} B dom {A} V)
3 elex 2542 . . . 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 3938 . . . . 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 2083 . . . . 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 4287 . . . 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 3360 . . . . . . . . . . . . 13 (A = ⟨x, y⟩ → {A} = {⟨x, y⟩})
1413rneqd 4488 . . . . . . . . . . . 12 (A = ⟨x, y⟩ → ran {A} = ran {⟨x, y⟩})
1514unieqd 3564 . . . . . . . . . . 11 (A = ⟨x, y⟩ → ran {A} = ran {⟨x, y⟩})
16 vex 2537 . . . . . . . . . . . 12 x V
17 vex 2537 . . . . . . . . . . . 12 y V
1816, 17op2nda 4730 . . . . . . . . . . 11 ran {⟨x, y⟩} = y
1915, 18syl6req 2072 . . . . . . . . . 10 (A = ⟨x, y⟩ → y = ran {A})
2019pm4.71ri 372 . . . . . . . . 9 (A = ⟨x, y⟩ ↔ (y = ran {A} A = ⟨x, y⟩))
2120anbi1i 434 . . . . . . . 8 ((A = ⟨x, y (x B y 𝐶)) ↔ ((y = ran {A} A = ⟨x, y⟩) (x B y 𝐶)))
22 anass 383 . . . . . . . 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 1480 . . . . . 6 (y(A = ⟨x, y (x B y 𝐶)) ↔ y(y = ran {A} (A = ⟨x, y (x B y 𝐶))))
25 snexgOLD 3908 . . . . . . . . 9 (A V → {A} V)
26 rnexg 4522 . . . . . . . . 9 ({A} V → ran {A} V)
2725, 26syl 14 . . . . . . . 8 (A V → ran {A} V)
28 uniexg 4123 . . . . . . . 8 (ran {A} V → ran {A} V)
2927, 28syl 14 . . . . . . 7 (A V → ran {A} V)
30 opeq2 3523 . . . . . . . . . 10 (y = ran {A} → ⟨x, y⟩ = ⟨x, ran {A}⟩)
3130eqeq2d 2034 . . . . . . . . 9 (y = ran {A} → (A = ⟨x, y⟩ ↔ A = ⟨x, ran {A}⟩))
32 eleq1 2083 . . . . . . . . . 10 (y = ran {A} → (y 𝐶 ran {A} 𝐶))
3332anbi2d 440 . . . . . . . . 9 (y = ran {A} → ((x B y 𝐶) ↔ (x B ran {A} 𝐶)))
3431, 33anbi12d 445 . . . . . . . 8 (y = ran {A} → ((A = ⟨x, y (x B y 𝐶)) ↔ (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))))
3534ceqsexgv 2649 . . . . . . 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 3360 . . . . . . . . . . . 12 (A = ⟨x, ran {A}⟩ → {A} = {⟨x, ran {A}⟩})
3938dmeqd 4462 . . . . . . . . . . 11 (A = ⟨x, ran {A}⟩ → dom {A} = dom {⟨x, ran {A}⟩})
4039unieqd 3564 . . . . . . . . . 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 4717 . . . . . . . . . . . . 13 ( ran {A} V → dom {⟨x, ran {A}⟩} = {x})
4329, 42syl 14 . . . . . . . . . . . 12 (A V → dom {⟨x, ran {A}⟩} = {x})
4443unieqd 3564 . . . . . . . . . . 11 (A V → dom {⟨x, ran {A}⟩} = {x})
4516unisn 3569 . . . . . . . . . . 11 {x} = x
4644, 45syl6eq 2071 . . . . . . . . . 10 (A V → dom {⟨x, ran {A}⟩} = x)
4746adantr 261 . . . . . . . . 9 ((A V A = ⟨x, ran {A}⟩) → dom {⟨x, ran {A}⟩} = x)
4841, 47eqtr2d 2056 . . . . . . . 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 441 . . . . 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 383 . . . . . 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 1689 . . 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 4521 . . . . . 6 ({A} V → dom {A} V)
5725, 56syl 14 . . . . 5 (A V → dom {A} V)
58 uniexg 4123 . . . . 5 (dom {A} V → dom {A} V)
5957, 58syl 14 . . . 4 (A V → dom {A} V)
60 opeq1 3522 . . . . . . 7 (x = dom {A} → ⟨x, ran {A}⟩ = ⟨ dom {A}, ran {A}⟩)
6160eqeq2d 2034 . . . . . 6 (x = dom {A} → (A = ⟨x, ran {A}⟩ ↔ A = ⟨ dom {A}, ran {A}⟩))
62 eleq1 2083 . . . . . . 7 (x = dom {A} → (x B dom {A} B))
6362anbi1d 441 . . . . . 6 (x = dom {A} → ((x B ran {A} 𝐶) ↔ ( dom {A} B ran {A} 𝐶)))
6461, 63anbi12d 445 . . . . 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 2649 . . . 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 607 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 1228  wex 1363   wcel 1375  Vcvv 2534  {csn 3349  cop 3352   cuni 3553   × cxp 4268  dom cdm 4270  ran crn 4271
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917  ax-un 4118
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358  df-uni 3554  df-br 3738  df-opab 3792  df-xp 4276  df-rel 4277  df-cnv 4278  df-dm 4280  df-rn 4281
This theorem is referenced by:  elxp6  5716
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