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Theorem dfco2a 4899
Description: Generalization of dfco2 4898, 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) = 𝑥𝐶 ((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 4898 . 2 (𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
2 vex 2618 . . . . . . . . . . . . . 14 𝑥 ∈ V
3 vex 2618 . . . . . . . . . . . . . . 15 𝑧 ∈ V
43eliniseg 4771 . . . . . . . . . . . . . 14 (𝑥 ∈ V → (𝑧 ∈ (𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥))
52, 4ax-mp 7 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥)
63, 2brelrn 4638 . . . . . . . . . . . . 13 (𝑧𝐵𝑥𝑥 ∈ ran 𝐵)
75, 6sylbi 119 . . . . . . . . . . . 12 (𝑧 ∈ (𝐵 “ {𝑥}) → 𝑥 ∈ ran 𝐵)
8 vex 2618 . . . . . . . . . . . . . 14 𝑤 ∈ V
92, 8elimasn 4768 . . . . . . . . . . . . 13 (𝑤 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐴)
102, 8opeldm 4609 . . . . . . . . . . . . 13 (⟨𝑥, 𝑤⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
119, 10sylbi 119 . . . . . . . . . . . 12 (𝑤 ∈ (𝐴 “ {𝑥}) → 𝑥 ∈ dom 𝐴)
127, 11anim12ci 332 . . . . . . . . . . 11 ((𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥})) → (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
1312adantl 271 . . . . . . . . . 10 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
1413exlimivv 1821 . . . . . . . . 9 (∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
15 elxp 4430 . . . . . . . . 9 (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))))
16 elin 3172 . . . . . . . . 9 (𝑥 ∈ (dom 𝐴 ∩ ran 𝐵) ↔ (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
1714, 15, 163imtr4i 199 . . . . . . . 8 (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) → 𝑥 ∈ (dom 𝐴 ∩ ran 𝐵))
18 ssel 3008 . . . . . . . 8 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑥 ∈ (dom 𝐴 ∩ ran 𝐵) → 𝑥𝐶))
1917, 18syl5 32 . . . . . . 7 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) → 𝑥𝐶))
2019pm4.71rd 386 . . . . . 6 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
2120exbidv 1750 . . . . 5 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
22 rexv 2631 . . . . 5 (∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23 df-rex 2361 . . . . 5 (∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
2421, 22, 233bitr4g 221 . . . 4 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
25 eliun 3719 . . . 4 (𝑦 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
26 eliun 3719 . . . 4 (𝑦 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
2724, 25, 263bitr4g 221 . . 3 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ 𝑦 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
2827eqrdv 2083 . 2 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) = 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
291, 28syl5eq 2129 1 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴𝐵) = 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wex 1424  wcel 1436  wrex 2356  Vcvv 2615  cin 2987  wss 2988  {csn 3431  cop 3434   ciun 3715   class class class wbr 3822   × cxp 4411  ccnv 4412  dom cdm 4413  ran crn 4414  cima 4416  ccom 4417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-iun 3717  df-br 3823  df-opab 3877  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426
This theorem is referenced by: (None)
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