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Theorem crossex 5850
 Description: The function mapping x and y to their cross product is a set. (Contributed by SF, 11-Feb-2015.)
Assertion
Ref Expression
crossex Cross V

Proof of Theorem crossex
Dummy variables a b x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cross 5764 . . 3 Cross = (x V, y V (x × y))
2 rexcom 2772 . . . . 5 (a x b y z = a, bb y a x z = a, b)
3 elxp2 4802 . . . . 5 (z (x × y) ↔ a x b y z = a, b)
4 elin 3219 . . . . . . . 8 ({b}, {z}, x, y ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)) ↔ ({b}, {z}, x, y Ins2 Ins2 S {b}, {z}, x, y Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)))
5 snex 4111 . . . . . . . . . . 11 {z} V
65otelins2 5791 . . . . . . . . . 10 ({b}, {z}, x, y Ins2 Ins2 S {b}, x, y Ins2 S )
7 vex 2862 . . . . . . . . . . 11 x V
87otelins2 5791 . . . . . . . . . 10 ({b}, x, y Ins2 S {b}, y S )
9 vex 2862 . . . . . . . . . . 11 b V
10 vex 2862 . . . . . . . . . . 11 y V
119, 10opelssetsn 4760 . . . . . . . . . 10 ({b}, y S b y)
126, 8, 113bitri 262 . . . . . . . . 9 ({b}, {z}, x, y Ins2 Ins2 S b y)
1310oqelins4 5794 . . . . . . . . . 10 ({b}, {z}, x, y Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c) ↔ {b}, {z}, x (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c))
14 elin 3219 . . . . . . . . . . . . 13 ({a}, {b}, {z}, x ( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) ↔ ({a}, {b}, {z}, x Ins2 Ins2 S {a}, {b}, {z}, x Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))))
15 snex 4111 . . . . . . . . . . . . . . . 16 {b} V
1615otelins2 5791 . . . . . . . . . . . . . . 15 ({a}, {b}, {z}, x Ins2 Ins2 S {a}, {z}, x Ins2 S )
175otelins2 5791 . . . . . . . . . . . . . . 15 ({a}, {z}, x Ins2 S {a}, x S )
18 vex 2862 . . . . . . . . . . . . . . . 16 a V
1918, 7opelssetsn 4760 . . . . . . . . . . . . . . 15 ({a}, x S a x)
2016, 17, 193bitri 262 . . . . . . . . . . . . . 14 ({a}, {b}, {z}, x Ins2 Ins2 S a x)
217oqelins4 5794 . . . . . . . . . . . . . . 15 ({a}, {b}, {z}, x Ins4 SI3 ( Ins2 1st ∩ (V × 2nd )) ↔ {a}, {b}, {z} SI3 ( Ins2 1st ∩ (V × 2nd )))
22 vex 2862 . . . . . . . . . . . . . . . 16 z V
2318, 9, 22otsnelsi3 5805 . . . . . . . . . . . . . . 15 ({a}, {b}, {z} SI3 ( Ins2 1st ∩ (V × 2nd )) ↔ a, b, z ( Ins2 1st ∩ (V × 2nd )))
24 elin 3219 . . . . . . . . . . . . . . . 16 (a, b, z ( Ins2 1st ∩ (V × 2nd )) ↔ (a, b, z Ins2 1st a, b, z (V × 2nd )))
259otelins2 5791 . . . . . . . . . . . . . . . . . 18 (a, b, z Ins2 1sta, z 1st )
26 df-br 4640 . . . . . . . . . . . . . . . . . 18 (a1st za, z 1st )
27 brcnv 4892 . . . . . . . . . . . . . . . . . 18 (a1st zz1st a)
2825, 26, 273bitr2i 264 . . . . . . . . . . . . . . . . 17 (a, b, z Ins2 1stz1st a)
29 opelxp 4811 . . . . . . . . . . . . . . . . . . 19 (a, b, z (V × 2nd ) ↔ (a V b, z 2nd ))
3018, 29mpbiran 884 . . . . . . . . . . . . . . . . . 18 (a, b, z (V × 2nd ) ↔ b, z 2nd )
31 df-br 4640 . . . . . . . . . . . . . . . . . 18 (b2nd zb, z 2nd )
32 brcnv 4892 . . . . . . . . . . . . . . . . . 18 (b2nd zz2nd b)
3330, 31, 323bitr2i 264 . . . . . . . . . . . . . . . . 17 (a, b, z (V × 2nd ) ↔ z2nd b)
3428, 33anbi12i 678 . . . . . . . . . . . . . . . 16 ((a, b, z Ins2 1st a, b, z (V × 2nd )) ↔ (z1st a z2nd b))
3518, 9op1st2nd 5790 . . . . . . . . . . . . . . . 16 ((z1st a z2nd b) ↔ z = a, b)
3624, 34, 353bitri 262 . . . . . . . . . . . . . . 15 (a, b, z ( Ins2 1st ∩ (V × 2nd )) ↔ z = a, b)
3721, 23, 363bitri 262 . . . . . . . . . . . . . 14 ({a}, {b}, {z}, x Ins4 SI3 ( Ins2 1st ∩ (V × 2nd )) ↔ z = a, b)
3820, 37anbi12i 678 . . . . . . . . . . . . 13 (({a}, {b}, {z}, x Ins2 Ins2 S {a}, {b}, {z}, x Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) ↔ (a x z = a, b))
3914, 38bitri 240 . . . . . . . . . . . 12 ({a}, {b}, {z}, x ( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) ↔ (a x z = a, b))
4039exbii 1582 . . . . . . . . . . 11 (a{a}, {b}, {z}, x ( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) ↔ a(a x z = a, b))
41 elima1c 4947 . . . . . . . . . . 11 ({b}, {z}, x (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c) ↔ a{a}, {b}, {z}, x ( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))))
42 df-rex 2620 . . . . . . . . . . 11 (a x z = a, ba(a x z = a, b))
4340, 41, 423bitr4i 268 . . . . . . . . . 10 ({b}, {z}, x (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c) ↔ a x z = a, b)
4413, 43bitri 240 . . . . . . . . 9 ({b}, {z}, x, y Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c) ↔ a x z = a, b)
4512, 44anbi12i 678 . . . . . . . 8 (({b}, {z}, x, y Ins2 Ins2 S {b}, {z}, x, y Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)) ↔ (b y a x z = a, b))
464, 45bitri 240 . . . . . . 7 ({b}, {z}, x, y ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)) ↔ (b y a x z = a, b))
4746exbii 1582 . . . . . 6 (b{b}, {z}, x, y ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)) ↔ b(b y a x z = a, b))
48 elima1c 4947 . . . . . 6 ({z}, x, y (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)) “ 1c) ↔ b{b}, {z}, x, y ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)))
49 df-rex 2620 . . . . . 6 (b y a x z = a, bb(b y a x z = a, b))
5047, 48, 493bitr4i 268 . . . . 5 ({z}, x, y (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)) “ 1c) ↔ b y a x z = a, b)
512, 3, 503bitr4ri 269 . . . 4 ({z}, x, y (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)) “ 1c) ↔ z (x × y))
5251releqmpt2 5809 . . 3 (((V × V) × V) (( Ins2 S Ins3 (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)) “ 1c)) “ 1c)) = (x V, y V (x × y))
531, 52eqtr4i 2376 . 2 Cross = (((V × V) × V) (( Ins2 S Ins3 (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)) “ 1c)) “ 1c))
54 vvex 4109 . . 3 V V
55 ssetex 4744 . . . . . . 7 S V
5655ins2ex 5797 . . . . . 6 Ins2 S V
5756ins2ex 5797 . . . . 5 Ins2 Ins2 S V
58 1stex 4739 . . . . . . . . . . . . 13 1st V
5958cnvex 5102 . . . . . . . . . . . 12 1st V
6059ins2ex 5797 . . . . . . . . . . 11 Ins2 1st V
61 2ndex 5112 . . . . . . . . . . . . 13 2nd V
6261cnvex 5102 . . . . . . . . . . . 12 2nd V
6354, 62xpex 5115 . . . . . . . . . . 11 (V × 2nd ) V
6460, 63inex 4105 . . . . . . . . . 10 ( Ins2 1st ∩ (V × 2nd )) V
6564si3ex 5806 . . . . . . . . 9 SI3 ( Ins2 1st ∩ (V × 2nd )) V
6665ins4ex 5799 . . . . . . . 8 Ins4 SI3 ( Ins2 1st ∩ (V × 2nd )) V
6757, 66inex 4105 . . . . . . 7 ( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) V
68 1cex 4142 . . . . . . 7 1c V
6967, 68imaex 4747 . . . . . 6 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c) V
7069ins4ex 5799 . . . . 5 Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c) V
7157, 70inex 4105 . . . 4 ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)) V
7271, 68imaex 4747 . . 3 (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)) “ 1c) V
7354, 54, 72mpt2exlem 5811 . 2 (((V × V) × V) (( Ins2 S Ins3 (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins2 1st ∩ (V × 2nd ))) “ 1c)) “ 1c)) “ 1c)) V
7453, 73eqeltri 2423 1 Cross V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859   ∖ cdif 3206   ∩ cin 3208   ⊕ csymdif 3209  {csn 3737  1cc1c 4134  ⟨cop 4561   class class class wbr 4639  1st c1st 4717   S csset 4719   “ cima 4722   × cxp 4770  ◡ccnv 4771  2nd c2nd 4783   ↦ cmpt2 5653   Ins2 cins2 5749   Ins3 cins3 5751   Ins4 cins4 5755   SI3 csi3 5757   Cross ccross 5763 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-xp 4784  df-cnv 4785  df-2nd 4797  df-oprab 5528  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-ins4 5756  df-si3 5758  df-cross 5764 This theorem is referenced by:  ovmuc  6130  mucex  6133
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