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Theorem dfco2a 5141
Description: Generalization of dfco2 5140, 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 5140 . 2 (𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
2 vex 2752 . . . . . . . . . . . . . 14 𝑥 ∈ V
3 vex 2752 . . . . . . . . . . . . . . 15 𝑧 ∈ V
43eliniseg 5010 . . . . . . . . . . . . . 14 (𝑥 ∈ V → (𝑧 ∈ (𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥))
52, 4ax-mp 5 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥)
63, 2brelrn 4872 . . . . . . . . . . . . 13 (𝑧𝐵𝑥𝑥 ∈ ran 𝐵)
75, 6sylbi 121 . . . . . . . . . . . 12 (𝑧 ∈ (𝐵 “ {𝑥}) → 𝑥 ∈ ran 𝐵)
8 vex 2752 . . . . . . . . . . . . . 14 𝑤 ∈ V
92, 8elimasn 5007 . . . . . . . . . . . . 13 (𝑤 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐴)
102, 8opeldm 4842 . . . . . . . . . . . . 13 (⟨𝑥, 𝑤⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
119, 10sylbi 121 . . . . . . . . . . . 12 (𝑤 ∈ (𝐴 “ {𝑥}) → 𝑥 ∈ dom 𝐴)
127, 11anim12ci 339 . . . . . . . . . . 11 ((𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥})) → (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
1312adantl 277 . . . . . . . . . 10 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
1413exlimivv 1906 . . . . . . . . 9 (∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
15 elxp 4655 . . . . . . . . 9 (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))))
16 elin 3330 . . . . . . . . 9 (𝑥 ∈ (dom 𝐴 ∩ ran 𝐵) ↔ (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
1714, 15, 163imtr4i 201 . . . . . . . 8 (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) → 𝑥 ∈ (dom 𝐴 ∩ ran 𝐵))
18 ssel 3161 . . . . . . . 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 394 . . . . . 6 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
2120exbidv 1835 . . . . 5 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
22 rexv 2767 . . . . 5 (∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23 df-rex 2471 . . . . 5 (∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
2421, 22, 233bitr4g 223 . . . 4 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
25 eliun 3902 . . . 4 (𝑦 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
26 eliun 3902 . . . 4 (𝑦 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
2724, 25, 263bitr4g 223 . . 3 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ 𝑦 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
2827eqrdv 2185 . 2 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) = 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
291, 28eqtrid 2232 1 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴𝐵) = 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wex 1502  wcel 2158  wrex 2466  Vcvv 2749  cin 3140  wss 3141  {csn 3604  cop 3607   ciun 3898   class class class wbr 4015   × cxp 4636  ccnv 4637  dom cdm 4638  ran crn 4639  cima 4641  ccom 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-iun 3900  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651
This theorem is referenced by: (None)
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