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Theorem dfco2a 6245
Description: Generalization of dfco2 6244, 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 6244 . 2 (𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
2 vex 3478 . . . . . . . . . . . . . . 15 𝑧 ∈ V
32eliniseg 6093 . . . . . . . . . . . . . 14 (𝑥 ∈ V → (𝑧 ∈ (𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥))
43elv 3480 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥)
5 vex 3478 . . . . . . . . . . . . . 14 𝑥 ∈ V
62, 5brelrn 5941 . . . . . . . . . . . . 13 (𝑧𝐵𝑥𝑥 ∈ ran 𝐵)
74, 6sylbi 216 . . . . . . . . . . . 12 (𝑧 ∈ (𝐵 “ {𝑥}) → 𝑥 ∈ ran 𝐵)
8 vex 3478 . . . . . . . . . . . . . 14 𝑤 ∈ V
95, 8elimasn 6088 . . . . . . . . . . . . 13 (𝑤 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐴)
105, 8opeldm 5907 . . . . . . . . . . . . 13 (⟨𝑥, 𝑤⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
119, 10sylbi 216 . . . . . . . . . . . 12 (𝑤 ∈ (𝐴 “ {𝑥}) → 𝑥 ∈ dom 𝐴)
127, 11anim12ci 614 . . . . . . . . . . 11 ((𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥})) → (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
1312adantl 482 . . . . . . . . . 10 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
1413exlimivv 1935 . . . . . . . . 9 (∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
15 elxp 5699 . . . . . . . . 9 (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))))
16 elin 3964 . . . . . . . . 9 (𝑥 ∈ (dom 𝐴 ∩ ran 𝐵) ↔ (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
1714, 15, 163imtr4i 291 . . . . . . . 8 (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) → 𝑥 ∈ (dom 𝐴 ∩ ran 𝐵))
18 ssel 3975 . . . . . . . 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 563 . . . . . 6 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
2120exbidv 1924 . . . . 5 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
22 rexv 3499 . . . . 5 (∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23 df-rex 3071 . . . . 5 (∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
2421, 22, 233bitr4g 313 . . . 4 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
25 eliun 5001 . . . 4 (𝑦 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
26 eliun 5001 . . . 4 (𝑦 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
2724, 25, 263bitr4g 313 . . 3 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ 𝑦 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
2827eqrdv 2730 . 2 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) = 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
291, 28eqtrid 2784 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 396   = wceq 1541  wex 1781  wcel 2106  wrex 3070  Vcvv 3474  cin 3947  wss 3948  {csn 4628  cop 4634   ciun 4997   class class class wbr 5148   × cxp 5674  ccnv 5675  dom cdm 5676  ran crn 5677  cima 5679  ccom 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-iun 4999  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  fparlem3  8099  fparlem4  8100
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