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Theorem dfco2a 4748
Description: Generalization of dfco2 4747, 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 4747 . 2 (AB) = x V ((B “ {x}) × (A “ {x}))
2 vex 2538 . . . . . . . . . . . . . 14 x V
3 vex 2538 . . . . . . . . . . . . . . 15 z V
43eliniseg 4622 . . . . . . . . . . . . . 14 (x V → (z (B “ {x}) ↔ zBx))
52, 4ax-mp 7 . . . . . . . . . . . . 13 (z (B “ {x}) ↔ zBx)
63, 2brelrn 4494 . . . . . . . . . . . . 13 (zBxx ran B)
75, 6sylbi 114 . . . . . . . . . . . 12 (z (B “ {x}) → x ran B)
8 vex 2538 . . . . . . . . . . . . . 14 w V
92, 8elimasn 4619 . . . . . . . . . . . . 13 (w (A “ {x}) ↔ ⟨x, w A)
102, 8opeldm 4465 . . . . . . . . . . . . 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 1758 . . . . . . . . 9 (zw(y = ⟨z, w (z (B “ {x}) w (A “ {x}))) → (x dom A x ran B))
15 elxp 4289 . . . . . . . . 9 (y ((B “ {x}) × (A “ {x})) ↔ zw(y = ⟨z, w (z (B “ {x}) w (A “ {x}))))
16 elin 3103 . . . . . . . . 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 2916 . . . . . . . 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 1688 . . . . 5 ((dom A ∩ ran B) ⊆ 𝐶 → (x y ((B “ {x}) × (A “ {x})) ↔ x(x 𝐶 y ((B “ {x}) × (A “ {x})))))
22 rexv 2549 . . . . 5 (x V y ((B “ {x}) × (A “ {x})) ↔ x y ((B “ {x}) × (A “ {x})))
23 df-rex 2290 . . . . 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 3635 . . . 4 (y x V ((B “ {x}) × (A “ {x})) ↔ x V y ((B “ {x}) × (A “ {x})))
26 eliun 3635 . . . 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 2020 . 2 ((dom A ∩ ran B) ⊆ 𝐶 x V ((B “ {x}) × (A “ {x})) = x 𝐶 ((B “ {x}) × (A “ {x})))
291, 28syl5eq 2066 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 1228  wex 1362   wcel 1374  wrex 2285  Vcvv 2535  cin 2893  wss 2894  {csn 3350  cop 3353   ciun 3631   class class class wbr 3738   × cxp 4270  ccnv 4271  dom cdm 4272  ran crn 4273  cima 4275  ccom 4276
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-iun 3633  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285
This theorem is referenced by: (None)
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