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Theorem dfco2a 4764
Description: Generalization of dfco2 4763, where 𝐶 can have any value between dom A ∩ ran B and V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dfco2a ((dom A ∩ ran B) ⊆ 𝐶 → (AB) = 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 4763 . 2 (AB) = x V ((B “ {x}) × (A “ {x}))
2 vex 2554 . . . . . . . . . . . . . 14 x V
3 vex 2554 . . . . . . . . . . . . . . 15 z V
43eliniseg 4638 . . . . . . . . . . . . . 14 (x V → (z (B “ {x}) ↔ zBx))
52, 4ax-mp 7 . . . . . . . . . . . . 13 (z (B “ {x}) ↔ zBx)
63, 2brelrn 4510 . . . . . . . . . . . . 13 (zBxx ran B)
75, 6sylbi 114 . . . . . . . . . . . 12 (z (B “ {x}) → x ran B)
8 vex 2554 . . . . . . . . . . . . . 14 w V
92, 8elimasn 4635 . . . . . . . . . . . . 13 (w (A “ {x}) ↔ ⟨x, w A)
102, 8opeldm 4481 . . . . . . . . . . . . 13 (⟨x, w Ax dom A)
119, 10sylbi 114 . . . . . . . . . . . 12 (w (A “ {x}) → x dom A)
127, 11anim12ci 322 . . . . . . . . . . 11 ((z (B “ {x}) w (A “ {x})) → (x dom A x ran B))
1312adantl 262 . . . . . . . . . 10 ((y = ⟨z, w (z (B “ {x}) w (A “ {x}))) → (x dom A x ran B))
1413exlimivv 1773 . . . . . . . . 9 (zw(y = ⟨z, w (z (B “ {x}) w (A “ {x}))) → (x dom A x ran B))
15 elxp 4305 . . . . . . . . 9 (y ((B “ {x}) × (A “ {x})) ↔ zw(y = ⟨z, w (z (B “ {x}) w (A “ {x}))))
16 elin 3120 . . . . . . . . 9 (x (dom A ∩ ran B) ↔ (x dom A x ran B))
1714, 15, 163imtr4i 190 . . . . . . . 8 (y ((B “ {x}) × (A “ {x})) → x (dom A ∩ ran B))
18 ssel 2933 . . . . . . . 8 ((dom A ∩ ran B) ⊆ 𝐶 → (x (dom A ∩ ran B) → x 𝐶))
1917, 18syl5 28 . . . . . . 7 ((dom A ∩ ran B) ⊆ 𝐶 → (y ((B “ {x}) × (A “ {x})) → x 𝐶))
2019pm4.71rd 374 . . . . . 6 ((dom A ∩ ran B) ⊆ 𝐶 → (y ((B “ {x}) × (A “ {x})) ↔ (x 𝐶 y ((B “ {x}) × (A “ {x})))))
2120exbidv 1703 . . . . 5 ((dom A ∩ ran B) ⊆ 𝐶 → (x y ((B “ {x}) × (A “ {x})) ↔ x(x 𝐶 y ((B “ {x}) × (A “ {x})))))
22 rexv 2566 . . . . 5 (x V y ((B “ {x}) × (A “ {x})) ↔ x y ((B “ {x}) × (A “ {x})))
23 df-rex 2306 . . . . 5 (x 𝐶 y ((B “ {x}) × (A “ {x})) ↔ x(x 𝐶 y ((B “ {x}) × (A “ {x}))))
2421, 22, 233bitr4g 212 . . . 4 ((dom A ∩ ran B) ⊆ 𝐶 → (x V y ((B “ {x}) × (A “ {x})) ↔ x 𝐶 y ((B “ {x}) × (A “ {x}))))
25 eliun 3652 . . . 4 (y x V ((B “ {x}) × (A “ {x})) ↔ x V y ((B “ {x}) × (A “ {x})))
26 eliun 3652 . . . 4 (y x 𝐶 ((B “ {x}) × (A “ {x})) ↔ x 𝐶 y ((B “ {x}) × (A “ {x})))
2724, 25, 263bitr4g 212 . . 3 ((dom A ∩ ran B) ⊆ 𝐶 → (y x V ((B “ {x}) × (A “ {x})) ↔ y x 𝐶 ((B “ {x}) × (A “ {x}))))
2827eqrdv 2035 . 2 ((dom A ∩ ran B) ⊆ 𝐶 x V ((B “ {x}) × (A “ {x})) = x 𝐶 ((B “ {x}) × (A “ {x})))
291, 28syl5eq 2081 1 ((dom A ∩ ran B) ⊆ 𝐶 → (AB) = x 𝐶 ((B “ {x}) × (A “ {x})))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  wrex 2301  Vcvv 2551  cin 2910  wss 2911  {csn 3367  cop 3370   ciun 3648   class class class wbr 3755   × cxp 4286  ccnv 4287  dom cdm 4288  ran crn 4289  cima 4291  ccom 4292
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-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
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-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by: (None)
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