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Theorem elxp4 4702
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4703. (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 2541 . 2 (A (B × 𝐶) → A V)
2 elex 2541 . . . 4 ( dom {A} B dom {A} V)
3 elex 2541 . . . 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 3917 . . . . 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 2082 . . . . 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 4255 . . . 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 3338 . . . . . . . . . . . . 13 (A = ⟨x, y⟩ → {A} = {⟨x, y⟩})
1413rneqd 4456 . . . . . . . . . . . 12 (A = ⟨x, y⟩ → ran {A} = ran {⟨x, y⟩})
1514unieqd 3543 . . . . . . . . . . 11 (A = ⟨x, y⟩ → ran {A} = ran {⟨x, y⟩})
16 vex 2536 . . . . . . . . . . . 12 x V
17 vex 2536 . . . . . . . . . . . 12 y V
1816, 17op2nda 4699 . . . . . . . . . . 11 ran {⟨x, y⟩} = y
1915, 18syl6req 2071 . . . . . . . . . 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 1479 . . . . . 6 (y(A = ⟨x, y (x B y 𝐶)) ↔ y(y = ran {A} (A = ⟨x, y (x B y 𝐶))))
25 snexgOLD 3887 . . . . . . . . 9 (A V → {A} V)
26 rnexg 4490 . . . . . . . . 9 ({A} V → ran {A} V)
2725, 26syl 14 . . . . . . . 8 (A V → ran {A} V)
28 uniexg 4098 . . . . . . . 8 (ran {A} V → ran {A} V)
2927, 28syl 14 . . . . . . 7 (A V → ran {A} V)
30 opeq2 3502 . . . . . . . . . 10 (y = ran {A} → ⟨x, y⟩ = ⟨x, ran {A}⟩)
3130eqeq2d 2033 . . . . . . . . 9 (y = ran {A} → (A = ⟨x, y⟩ ↔ A = ⟨x, ran {A}⟩))
32 eleq1 2082 . . . . . . . . . 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 2648 . . . . . . 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 3338 . . . . . . . . . . . 12 (A = ⟨x, ran {A}⟩ → {A} = {⟨x, ran {A}⟩})
3938dmeqd 4430 . . . . . . . . . . 11 (A = ⟨x, ran {A}⟩ → dom {A} = dom {⟨x, ran {A}⟩})
4039unieqd 3543 . . . . . . . . . 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 4686 . . . . . . . . . . . . 13 ( ran {A} V → dom {⟨x, ran {A}⟩} = {x})
4329, 42syl 14 . . . . . . . . . . . 12 (A V → dom {⟨x, ran {A}⟩} = {x})
4443unieqd 3543 . . . . . . . . . . 11 (A V → dom {⟨x, ran {A}⟩} = {x})
4516unisn 3548 . . . . . . . . . . 11 {x} = x
4644, 45syl6eq 2070 . . . . . . . . . 10 (A V → dom {⟨x, ran {A}⟩} = x)
4746adantr 261 . . . . . . . . 9 ((A V A = ⟨x, ran {A}⟩) → dom {⟨x, ran {A}⟩} = x)
4841, 47eqtr2d 2055 . . . . . . . 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 1688 . . 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 4489 . . . . . 6 ({A} V → dom {A} V)
5725, 56syl 14 . . . . 5 (A V → dom {A} V)
58 uniexg 4098 . . . . 5 (dom {A} V → dom {A} V)
5957, 58syl 14 . . . 4 (A V → dom {A} V)
60 opeq1 3501 . . . . . . 7 (x = dom {A} → ⟨x, ran {A}⟩ = ⟨ dom {A}, ran {A}⟩)
6160eqeq2d 2033 . . . . . 6 (x = dom {A} → (A = ⟨x, ran {A}⟩ ↔ A = ⟨ dom {A}, ran {A}⟩))
62 eleq1 2082 . . . . . . 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 2648 . . . 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  wex 1361   = wceq 1373   wcel 1375  Vcvv 2533  {csn 3327  cop 3330   cuni 3532   × cxp 4236  dom cdm 4238  ran crn 4239
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896  ax-un 4093
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-br 3717  df-opab 3771  df-xp 4244  df-rel 4245  df-cnv 4246  df-dm 4248  df-rn 4249
This theorem is referenced by:  elxp6  5685
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