Theorem dfco2a 6150

Description: Generalization of dfco2 6149, 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 → (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 ((◡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.

Step	Hyp	Ref	Expression
1		dfco2 6149	. 2 ⊢ (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
2		vex 3436	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
3	2	eliniseg 6002	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ V → (𝑧 ∈ (◡𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥))
4	3	elv 3438	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (◡𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥)
5		vex 3436	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
6	2, 5	brelrn 5851	. . . . . . . . . . . . 13 ⊢ (𝑧𝐵𝑥 → 𝑥 ∈ ran 𝐵)
7	4, 6	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (◡𝐵 “ {𝑥}) → 𝑥 ∈ ran 𝐵)
8		vex 3436	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
9	5, 8	elimasn 5997	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐴)
10	5, 8	opeldm 5816	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑤⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
11	9, 10	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (𝐴 “ {𝑥}) → 𝑥 ∈ dom 𝐴)
12	7, 11	anim12ci 614	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥})) → (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
13	12	adantl 482	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
14	13	exlimivv 1935	. . . . . . . . 9 ⊢ (∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
15		elxp 5612	. . . . . . . . 9 ⊢ (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))))
16		elin 3903	. . . . . . . . 9 ⊢ (𝑥 ∈ (dom 𝐴 ∩ ran 𝐵) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
17	14, 15, 16	3imtr4i 292	. . . . . . . 8 ⊢ (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) → 𝑥 ∈ (dom 𝐴 ∩ ran 𝐵))
18		ssel 3914	. . . . . . . 8 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑥 ∈ (dom 𝐴 ∩ ran 𝐵) → 𝑥 ∈ 𝐶))
19	17, 18	syl5 34	. . . . . . 7 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) → 𝑥 ∈ 𝐶))
20	19	pm4.71rd 563	. . . . . 6 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
21	20	exbidv 1924	. . . . 5 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
22		rexv 3457	. . . . 5 ⊢ (∃𝑥 ∈ V 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23		df-rex 3070	. . . . 5 ⊢ (∃𝑥 ∈ 𝐶 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
24	21, 22, 23	3bitr4g 314	. . . 4 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 ∈ V 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ 𝐶 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
25		eliun 4928	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
26		eliun 4928	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ 𝐶 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
27	24, 25, 26	3bitr4g 314	. . 3 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
28	27	eqrdv 2736	. 2 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
29	1, 28	eqtrid 2790	1 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  {csn 4561  ⟨cop 4567  ∪ ciun 4924   class class class wbr 5074   × cxp 5587  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   “ cima 5592   ∘ ccom 5593

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-iun 4926  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602

This theorem is referenced by:  fparlem3  7954  fparlem4  7955