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Theorem elxp5 4752
Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4751 when the double intersection does not create class existence problems (caused by int0 3620). (Contributed by NM, 1-Aug-2004.)
Assertion
Ref Expression
elxp5 (A (B × 𝐶) ↔ (A = ⟨ A, ran {A}⟩ ( A B ran {A} 𝐶)))

Proof of Theorem elxp5
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 ( A B A V)
3 elex 2560 . . . 4 ( ran {A} 𝐶 ran {A} V)
42, 3anim12i 321 . . 3 (( A B ran {A} 𝐶) → ( A V ran {A} V))
5 opexgOLD 3956 . . . . 5 (( A V ran {A} V) → ⟨ A, ran {A}⟩ V)
65adantl 262 . . . 4 ((A = ⟨ A, ran {A}⟩ ( A V ran {A} V)) → ⟨ A, ran {A}⟩ V)
7 eleq1 2097 . . . . 5 (A = ⟨ A, ran {A}⟩ → (A V ↔ ⟨ A, ran {A}⟩ V))
87adantr 261 . . . 4 ((A = ⟨ A, ran {A}⟩ ( A V ran {A} V)) → (A V ↔ ⟨ A, ran {A}⟩ V))
96, 8mpbird 156 . . 3 ((A = ⟨ A, ran {A}⟩ ( A V ran {A} V)) → A V)
104, 9sylan2 270 . 2 ((A = ⟨ A, ran {A}⟩ ( A B ran {A} 𝐶)) → A V)
11 elxp 4305 . . . 4 (A (B × 𝐶) ↔ xy(A = ⟨x, y (x B y 𝐶)))
12 sneq 3378 . . . . . . . . . . . . . 14 (A = ⟨x, y⟩ → {A} = {⟨x, y⟩})
1312rneqd 4506 . . . . . . . . . . . . 13 (A = ⟨x, y⟩ → ran {A} = ran {⟨x, y⟩})
1413unieqd 3582 . . . . . . . . . . . 12 (A = ⟨x, y⟩ → ran {A} = ran {⟨x, y⟩})
15 vex 2554 . . . . . . . . . . . . 13 x V
16 vex 2554 . . . . . . . . . . . . 13 y V
1715, 16op2nda 4748 . . . . . . . . . . . 12 ran {⟨x, y⟩} = y
1814, 17syl6req 2086 . . . . . . . . . . 11 (A = ⟨x, y⟩ → y = ran {A})
1918pm4.71ri 372 . . . . . . . . . 10 (A = ⟨x, y⟩ ↔ (y = ran {A} A = ⟨x, y⟩))
2019anbi1i 431 . . . . . . . . 9 ((A = ⟨x, y (x B y 𝐶)) ↔ ((y = ran {A} A = ⟨x, y⟩) (x B y 𝐶)))
21 anass 381 . . . . . . . . 9 (((y = ran {A} A = ⟨x, y⟩) (x B y 𝐶)) ↔ (y = ran {A} (A = ⟨x, y (x B y 𝐶))))
2220, 21bitri 173 . . . . . . . 8 ((A = ⟨x, y (x B y 𝐶)) ↔ (y = ran {A} (A = ⟨x, y (x B y 𝐶))))
2322exbii 1493 . . . . . . 7 (y(A = ⟨x, y (x B y 𝐶)) ↔ y(y = ran {A} (A = ⟨x, y (x B y 𝐶))))
24 snexgOLD 3926 . . . . . . . . . 10 (A V → {A} V)
25 rnexg 4540 . . . . . . . . . 10 ({A} V → ran {A} V)
2624, 25syl 14 . . . . . . . . 9 (A V → ran {A} V)
27 uniexg 4141 . . . . . . . . 9 (ran {A} V → ran {A} V)
2826, 27syl 14 . . . . . . . 8 (A V → ran {A} V)
29 opeq2 3541 . . . . . . . . . . 11 (y = ran {A} → ⟨x, y⟩ = ⟨x, ran {A}⟩)
3029eqeq2d 2048 . . . . . . . . . 10 (y = ran {A} → (A = ⟨x, y⟩ ↔ A = ⟨x, ran {A}⟩))
31 eleq1 2097 . . . . . . . . . . 11 (y = ran {A} → (y 𝐶 ran {A} 𝐶))
3231anbi2d 437 . . . . . . . . . 10 (y = ran {A} → ((x B y 𝐶) ↔ (x B ran {A} 𝐶)))
3330, 32anbi12d 442 . . . . . . . . 9 (y = ran {A} → ((A = ⟨x, y (x B y 𝐶)) ↔ (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))))
3433ceqsexgv 2667 . . . . . . . 8 ( ran {A} V → (y(y = ran {A} (A = ⟨x, y (x B y 𝐶))) ↔ (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))))
3528, 34syl 14 . . . . . . 7 (A V → (y(y = ran {A} (A = ⟨x, y (x B y 𝐶))) ↔ (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))))
3623, 35syl5bb 181 . . . . . 6 (A V → (y(A = ⟨x, y (x B y 𝐶)) ↔ (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))))
37 inteq 3609 . . . . . . . . . . . 12 (A = ⟨x, ran {A}⟩ → A = x, ran {A}⟩)
3837inteqd 3611 . . . . . . . . . . 11 (A = ⟨x, ran {A}⟩ → A = x, ran {A}⟩)
3938adantl 262 . . . . . . . . . 10 ((A V A = ⟨x, ran {A}⟩) → A = x, ran {A}⟩)
40 op1stbg 4176 . . . . . . . . . . . 12 ((x V ran {A} V) → x, ran {A}⟩ = x)
4115, 28, 40sylancr 393 . . . . . . . . . . 11 (A V → x, ran {A}⟩ = x)
4241adantr 261 . . . . . . . . . 10 ((A V A = ⟨x, ran {A}⟩) → x, ran {A}⟩ = x)
4339, 42eqtr2d 2070 . . . . . . . . 9 ((A V A = ⟨x, ran {A}⟩) → x = A)
4443ex 108 . . . . . . . 8 (A V → (A = ⟨x, ran {A}⟩ → x = A))
4544pm4.71rd 374 . . . . . . 7 (A V → (A = ⟨x, ran {A}⟩ ↔ (x = A A = ⟨x, ran {A}⟩)))
4645anbi1d 438 . . . . . 6 (A V → ((A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶)) ↔ ((x = A A = ⟨x, ran {A}⟩) (x B ran {A} 𝐶))))
47 anass 381 . . . . . . 7 (((x = A A = ⟨x, ran {A}⟩) (x B ran {A} 𝐶)) ↔ (x = A (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))))
4847a1i 9 . . . . . 6 (A V → (((x = A A = ⟨x, ran {A}⟩) (x B ran {A} 𝐶)) ↔ (x = A (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶)))))
4936, 46, 483bitrd 203 . . . . 5 (A V → (y(A = ⟨x, y (x B y 𝐶)) ↔ (x = A (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶)))))
5049exbidv 1703 . . . 4 (A V → (xy(A = ⟨x, y (x B y 𝐶)) ↔ x(x = A (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶)))))
5111, 50syl5bb 181 . . 3 (A V → (A (B × 𝐶) ↔ x(x = A (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶)))))
52 eleq1 2097 . . . . . . 7 (x = A → (x V ↔ A V))
5315, 52mpbii 136 . . . . . 6 (x = A A V)
5453adantr 261 . . . . 5 ((x = A (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))) → A V)
5554exlimiv 1486 . . . 4 (x(x = A (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))) → A V)
562ad2antrl 459 . . . 4 ((A = ⟨ A, ran {A}⟩ ( A B ran {A} 𝐶)) → A V)
57 opeq1 3540 . . . . . . 7 (x = A → ⟨x, ran {A}⟩ = ⟨ A, ran {A}⟩)
5857eqeq2d 2048 . . . . . 6 (x = A → (A = ⟨x, ran {A}⟩ ↔ A = ⟨ A, ran {A}⟩))
59 eleq1 2097 . . . . . . 7 (x = A → (x B A B))
6059anbi1d 438 . . . . . 6 (x = A → ((x B ran {A} 𝐶) ↔ ( A B ran {A} 𝐶)))
6158, 60anbi12d 442 . . . . 5 (x = A → ((A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶)) ↔ (A = ⟨ A, ran {A}⟩ ( A B ran {A} 𝐶))))
6261ceqsexgv 2667 . . . 4 ( A V → (x(x = A (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))) ↔ (A = ⟨ A, ran {A}⟩ ( A B ran {A} 𝐶))))
6355, 56, 62pm5.21nii 619 . . 3 (x(x = A (A = ⟨x, ran {A}⟩ (x B ran {A} 𝐶))) ↔ (A = ⟨ A, ran {A}⟩ ( A B ran {A} 𝐶)))
6451, 63syl6bb 185 . 2 (A V → (A (B × 𝐶) ↔ (A = ⟨ A, ran {A}⟩ ( A B ran {A} 𝐶))))
651, 10, 64pm5.21nii 619 1 (A (B × 𝐶) ↔ (A = ⟨ A, ran {A}⟩ ( 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   cint 3606   × cxp 4286  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-int 3607  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: (None)
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