Theorem dfco2a 6193

Description: Generalization of dfco2 6192, 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 6192	. 2 ⊢ (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
2		vex 3440	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
3	2	eliniseg 6043	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ V → (𝑧 ∈ (◡𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥))
4	3	elv 3441	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (◡𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥)
5		vex 3440	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
6	2, 5	brelrn 5882	. . . . . . . . . . . . 13 ⊢ (𝑧𝐵𝑥 → 𝑥 ∈ ran 𝐵)
7	4, 6	sylbi 217	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (◡𝐵 “ {𝑥}) → 𝑥 ∈ ran 𝐵)
8		vex 3440	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
9	5, 8	elimasn 6039	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐴)
10	5, 8	opeldm 5847	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑤⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
11	9, 10	sylbi 217	. . . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
14	13	exlimivv 1933	. . . . . . . . 9 ⊢ (∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
15		elxp 5639	. . . . . . . . 9 ⊢ (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))))
16		elin 3918	. . . . . . . . 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 3928	. . . . . . . 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 562	. . . . . 6 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
21	20	exbidv 1922	. . . . 5 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
22		rexv 3464	. . . . 5 ⊢ (∃𝑥 ∈ V 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23		df-rex 3057	. . . . 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 4945	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
26		eliun 4945	. . . 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 2729	. 2 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
29	1, 28	eqtrid 2778	1 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436   ∩ cin 3901   ⊆ wss 3902  {csn 4576  ⟨cop 4582  ∪ ciun 4941   class class class wbr 5091   × cxp 5614  ^◡ccnv 5615  dom cdm 5616  ran crn 5617   “ cima 5619   ∘ ccom 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-iun 4943  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629

This theorem is referenced by:  fparlem3  8044  fparlem4  8045