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Theorem dfco2a 5594
Description: Generalization of dfco2 5593, 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 5593 . 2 (𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
2 vex 3189 . . . . . . . . . . . . . 14 𝑥 ∈ V
3 vex 3189 . . . . . . . . . . . . . . 15 𝑧 ∈ V
43eliniseg 5453 . . . . . . . . . . . . . 14 (𝑥 ∈ V → (𝑧 ∈ (𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥))
52, 4ax-mp 5 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥)
63, 2brelrn 5316 . . . . . . . . . . . . 13 (𝑧𝐵𝑥𝑥 ∈ ran 𝐵)
75, 6sylbi 207 . . . . . . . . . . . 12 (𝑧 ∈ (𝐵 “ {𝑥}) → 𝑥 ∈ ran 𝐵)
8 vex 3189 . . . . . . . . . . . . . 14 𝑤 ∈ V
92, 8elimasn 5449 . . . . . . . . . . . . 13 (𝑤 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐴)
102, 8opeldm 5288 . . . . . . . . . . . . 13 (⟨𝑥, 𝑤⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
119, 10sylbi 207 . . . . . . . . . . . 12 (𝑤 ∈ (𝐴 “ {𝑥}) → 𝑥 ∈ dom 𝐴)
127, 11anim12ci 590 . . . . . . . . . . 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 1857 . . . . . . . . 9 (∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
15 elxp 5091 . . . . . . . . 9 (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))))
16 elin 3774 . . . . . . . . 9 (𝑥 ∈ (dom 𝐴 ∩ ran 𝐵) ↔ (𝑥 ∈ dom 𝐴𝑥 ∈ ran 𝐵))
1714, 15, 163imtr4i 281 . . . . . . . 8 (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) → 𝑥 ∈ (dom 𝐴 ∩ ran 𝐵))
18 ssel 3577 . . . . . . . 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 666 . . . . . 6 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
2120exbidv 1847 . . . . 5 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
22 rexv 3206 . . . . 5 (∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23 df-rex 2913 . . . . 5 (∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥𝐶𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
2421, 22, 233bitr4g 303 . . . 4 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
25 eliun 4490 . . . 4 (𝑦 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
26 eliun 4490 . . . 4 (𝑦 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥𝐶 𝑦 ∈ ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
2724, 25, 263bitr4g 303 . . 3 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ 𝑦 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
2827eqrdv 2619 . 2 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) = 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
291, 28syl5eq 2667 1 ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴𝐵) = 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  Vcvv 3186  cin 3554  wss 3555  {csn 4148  cop 4154   ciun 4485   class class class wbr 4613   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  ccom 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-iun 4487  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087
This theorem is referenced by:  fparlem3  7224  fparlem4  7225
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