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Theorem dfco2a 5081
 Description: Generalization of dfco2 5080, where C can have any value between dom A ∩ ran B and V. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 21-Dec-2008.) (Revised by set.mm contributors, 27-Aug-2011.)
Assertion
Ref Expression
dfco2a ((dom A ∩ ran B) C → (A B) = x C ((B “ {x}) × (A “ {x})))
Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem dfco2a
Dummy variables w y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfco2 5080 . 2 (A B) = x V ((B “ {x}) × (A “ {x}))
2 elimasn 5019 . . . . . . . . . . . . 13 (w (A “ {x}) ↔ x, w A)
3 opeldm 4910 . . . . . . . . . . . . 13 (x, w Ax dom A)
42, 3sylbi 187 . . . . . . . . . . . 12 (w (A “ {x}) → x dom A)
5 eliniseg 5020 . . . . . . . . . . . . 13 (z (B “ {x}) ↔ zBx)
6 brelrn 4960 . . . . . . . . . . . . 13 (zBxx ran B)
75, 6sylbi 187 . . . . . . . . . . . 12 (z (B “ {x}) → x ran B)
84, 7anim12i 549 . . . . . . . . . . 11 ((w (A “ {x}) z (B “ {x})) → (x dom A x ran B))
98ancoms 439 . . . . . . . . . 10 ((z (B “ {x}) w (A “ {x})) → (x dom A x ran B))
109adantl 452 . . . . . . . . 9 ((y = z, w (z (B “ {x}) w (A “ {x}))) → (x dom A x ran B))
1110exlimivv 1635 . . . . . . . 8 (zw(y = z, w (z (B “ {x}) w (A “ {x}))) → (x dom A x ran B))
12 elxp 4801 . . . . . . . 8 (y ((B “ {x}) × (A “ {x})) ↔ zw(y = z, w (z (B “ {x}) w (A “ {x}))))
13 elin 3219 . . . . . . . 8 (x (dom A ∩ ran B) ↔ (x dom A x ran B))
1411, 12, 133imtr4i 257 . . . . . . 7 (y ((B “ {x}) × (A “ {x})) → x (dom A ∩ ran B))
15 ssel 3267 . . . . . . 7 ((dom A ∩ ran B) C → (x (dom A ∩ ran B) → x C))
1614, 15syl5 28 . . . . . 6 ((dom A ∩ ran B) C → (y ((B “ {x}) × (A “ {x})) → x C))
1716pm4.71rd 616 . . . . 5 ((dom A ∩ ran B) C → (y ((B “ {x}) × (A “ {x})) ↔ (x C y ((B “ {x}) × (A “ {x})))))
1817exbidv 1626 . . . 4 ((dom A ∩ ran B) C → (x y ((B “ {x}) × (A “ {x})) ↔ x(x C y ((B “ {x}) × (A “ {x})))))
19 eliun 3973 . . . . 5 (y x V ((B “ {x}) × (A “ {x})) ↔ x V y ((B “ {x}) × (A “ {x})))
20 rexv 2873 . . . . 5 (x V y ((B “ {x}) × (A “ {x})) ↔ x y ((B “ {x}) × (A “ {x})))
2119, 20bitri 240 . . . 4 (y x V ((B “ {x}) × (A “ {x})) ↔ x y ((B “ {x}) × (A “ {x})))
22 eliun 3973 . . . . 5 (y x C ((B “ {x}) × (A “ {x})) ↔ x C y ((B “ {x}) × (A “ {x})))
23 df-rex 2620 . . . . 5 (x C y ((B “ {x}) × (A “ {x})) ↔ x(x C y ((B “ {x}) × (A “ {x}))))
2422, 23bitri 240 . . . 4 (y x C ((B “ {x}) × (A “ {x})) ↔ x(x C y ((B “ {x}) × (A “ {x}))))
2518, 21, 243bitr4g 279 . . 3 ((dom A ∩ ran B) C → (y x V ((B “ {x}) × (A “ {x})) ↔ y x C ((B “ {x}) × (A “ {x}))))
2625eqrdv 2351 . 2 ((dom A ∩ ran B) Cx V ((B “ {x}) × (A “ {x})) = x C ((B “ {x}) × (A “ {x})))
271, 26syl5eq 2397 1 ((dom A ∩ ran B) C → (A B) = x C ((B “ {x}) × (A “ {x})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859   ∩ cin 3208   ⊆ wss 3257  {csn 3737  ∪ciun 3969  ⟨cop 4561   class class class wbr 4639   ∘ ccom 4721   “ cima 4722   × cxp 4770  ◡ccnv 4771  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-iun 3971  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-co 4726  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788 This theorem is referenced by: (None)
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