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Theorem dfco2a 6246
Description: Generalization of dfco2 6245, 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 6245 . 2 (𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
2 vex 3479 . . . . . . . . . . . . . . 15 𝑧 ∈ V
32eliniseg 6094 . . . . . . . . . . . . . 14 (𝑥 ∈ V → (𝑧 ∈ (𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥))
43elv 3481 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥)
5 vex 3479 . . . . . . . . . . . . . 14 𝑥 ∈ V
62, 5brelrn 5942 . . . . . . . . . . . . 13 (𝑧𝐵𝑥𝑥 ∈ ran 𝐵)
74, 6sylbi 216 . . . . . . . . . . . 12 (𝑧 ∈ (𝐵 “ {𝑥}) → 𝑥 ∈ ran 𝐵)
8 vex 3479 . . . . . . . . . . . . . 14 𝑤 ∈ V
95, 8elimasn 6089 . . . . . . . . . . . . 13 (𝑤 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐴)
105, 8opeldm 5908 . . . . . . . . . . . . 13 (⟨𝑥, 𝑤⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
119, 10sylbi 216 . . . . . . . . . . . 12 (𝑤 ∈ (𝐴 “ {𝑥}) → 𝑥 ∈ dom 𝐴)
127, 11anim12ci 615 . . . . . . . . . . 11 ((𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥})) → (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
1312adantl 483 . . . . . . . . . 10 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
1413exlimivv 1936 . . . . . . . . 9 (∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
15 elxp 5700 . . . . . . . . 9 (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))))
16 elin 3965 . . . . . . . . 9 (𝑥 ∈ (dom 𝐴 ∩ ran 𝐵) ↔ (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
1714, 15, 163imtr4i 292 . . . . . . . 8 (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) → 𝑥 ∈ (dom 𝐴 ∩ ran 𝐵))
18 ssel 3976 . . . . . . . 8 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑥 ∈ (dom 𝐴 ∩ ran 𝐵) → 𝑥𝐶))
1917, 18syl5 34 . . . . . . 7 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) → 𝑥𝐶))
2019pm4.71rd 564 . . . . . 6 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
2120exbidv 1925 . . . . 5 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
22 rexv 3500 . . . . 5 (∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23 df-rex 3072 . . . . 5 (∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
2421, 22, 233bitr4g 314 . . . 4 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
25 eliun 5002 . . . 4 (𝑦 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
26 eliun 5002 . . . 4 (𝑦 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
2724, 25, 263bitr4g 314 . . 3 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ 𝑦 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
2827eqrdv 2731 . 2 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) = 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
291, 28eqtrid 2785 1 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴𝐵) = 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wrex 3071  Vcvv 3475  cin 3948  wss 3949  {csn 4629  cop 4635   ciun 4998   class class class wbr 5149   × cxp 5675  ccnv 5676  dom cdm 5677  ran crn 5678  cima 5680  ccom 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iun 5000  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690
This theorem is referenced by:  fparlem3  8100  fparlem4  8101
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