New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  elxp4 GIF version

Theorem elxp4 5108
 Description: Membership in a cross product. This version requires no quantifiers or dummy variables. (Contributed by set.mm contributors, 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 elxp 4801 . 2 (A (B × C) ↔ xy(A = x, y (x B y C)))
2 sneq 3744 . . . . . . . . . . 11 (A = x, y → {A} = {x, y})
32rneqd 4958 . . . . . . . . . 10 (A = x, y → ran {A} = ran {x, y})
43unieqd 3902 . . . . . . . . 9 (A = x, yran {A} = ran {x, y})
5 vex 2862 . . . . . . . . . 10 x V
6 vex 2862 . . . . . . . . . 10 y V
75, 6op2nda 5076 . . . . . . . . 9 ran {x, y} = y
84, 7syl6req 2402 . . . . . . . 8 (A = x, yy = ran {A})
98adantr 451 . . . . . . 7 ((A = x, y (x B y C)) → y = ran {A})
109pm4.71ri 614 . . . . . 6 ((A = x, y (x B y C)) ↔ (y = ran {A} (A = x, y (x B y C))))
1110exbii 1582 . . . . 5 (y(A = x, y (x B y C)) ↔ y(y = ran {A} (A = x, y (x B y C))))
12 snex 4111 . . . . . . . 8 {A} V
1312rnex 5107 . . . . . . 7 ran {A} V
1413uniex 4317 . . . . . 6 ran {A} V
15 opeq2 4579 . . . . . . . 8 (y = ran {A} → x, y = x, ran {A})
1615eqeq2d 2364 . . . . . . 7 (y = ran {A} → (A = x, yA = x, ran {A}))
17 eleq1 2413 . . . . . . . 8 (y = ran {A} → (y Cran {A} C))
1817anbi2d 684 . . . . . . 7 (y = ran {A} → ((x B y C) ↔ (x B ran {A} C)))
1916, 18anbi12d 691 . . . . . 6 (y = ran {A} → ((A = x, y (x B y C)) ↔ (A = x, ran {A} (x B ran {A} C))))
2014, 19ceqsexv 2894 . . . . 5 (y(y = ran {A} (A = x, y (x B y C))) ↔ (A = x, ran {A} (x B ran {A} C)))
2111, 20bitri 240 . . . 4 (y(A = x, y (x B y C)) ↔ (A = x, ran {A} (x B ran {A} C)))
22 sneq 3744 . . . . . . . . 9 (A = x, ran {A} → {A} = {x, ran {A}})
2322dmeqd 4909 . . . . . . . 8 (A = x, ran {A} → dom {A} = dom {x, ran {A}})
2423unieqd 3902 . . . . . . 7 (A = x, ran {A}dom {A} = dom {x, ran {A}})
255, 14op1sta 5072 . . . . . . 7 dom {x, ran {A}} = x
2624, 25syl6req 2402 . . . . . 6 (A = x, ran {A}x = dom {A})
2726pm4.71ri 614 . . . . 5 (A = x, ran {A} ↔ (x = dom {A} A = x, ran {A}))
2827anbi1i 676 . . . 4 ((A = x, ran {A} (x B ran {A} C)) ↔ ((x = dom {A} A = x, ran {A}) (x B ran {A} C)))
29 anass 630 . . . 4 (((x = dom {A} A = x, ran {A}) (x B ran {A} C)) ↔ (x = dom {A} (A = x, ran {A} (x B ran {A} C))))
3021, 28, 293bitri 262 . . 3 (y(A = x, y (x B y C)) ↔ (x = dom {A} (A = x, ran {A} (x B ran {A} C))))
3130exbii 1582 . 2 (xy(A = x, y (x B y C)) ↔ x(x = dom {A} (A = x, ran {A} (x B ran {A} C))))
3212dmex 5106 . . . 4 dom {A} V
3332uniex 4317 . . 3 dom {A} V
34 opeq1 4578 . . . . 5 (x = dom {A} → x, ran {A} = dom {A}, ran {A})
3534eqeq2d 2364 . . . 4 (x = dom {A} → (A = x, ran {A}A = dom {A}, ran {A}))
36 eleq1 2413 . . . . 5 (x = dom {A} → (x Bdom {A} B))
3736anbi1d 685 . . . 4 (x = dom {A} → ((x B ran {A} C) ↔ (dom {A} B ran {A} C)))
3835, 37anbi12d 691 . . 3 (x = dom {A} → ((A = x, ran {A} (x B ran {A} C)) ↔ (A = dom {A}, ran {A} (dom {A} B ran {A} C))))
3933, 38ceqsexv 2894 . 2 (x(x = dom {A} (A = x, ran {A} (x B ran {A} C))) ↔ (A = dom {A}, ran {A} (dom {A} B ran {A} C)))
401, 31, 393bitri 262 1 (A (B × C) ↔ (A = dom {A}, ran {A} (dom {A} B ran {A} C)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {csn 3737  ∪cuni 3891  ⟨cop 4561   × cxp 4770  dom cdm 4772  ran crn 4773 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-swap 4724  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator