Theorem dfco2a 6244

Description: Generalization of dfco2 6243, 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 6243	. 2 ⊢ (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
2		vex 3473	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
3	2	eliniseg 6092	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ V → (𝑧 ∈ (◡𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥))
4	3	elv 3475	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (◡𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥)
5		vex 3473	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
6	2, 5	brelrn 5938	. . . . . . . . . . . . 13 ⊢ (𝑧𝐵𝑥 → 𝑥 ∈ ran 𝐵)
7	4, 6	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (◡𝐵 “ {𝑥}) → 𝑥 ∈ ran 𝐵)
8		vex 3473	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
9	5, 8	elimasn 6087	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐴)
10	5, 8	opeldm 5904	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑤⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
11	9, 10	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (𝐴 “ {𝑥}) → 𝑥 ∈ dom 𝐴)
12	7, 11	anim12ci 613	. . . . . . . . . . 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 1928	. . . . . . . . 9 ⊢ (∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
15		elxp 5695	. . . . . . . . 9 ⊢ (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))))
16		elin 3960	. . . . . . . . 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 3971	. . . . . . . 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 1917	. . . . 5 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
22		rexv 3495	. . . . 5 ⊢ (∃𝑥 ∈ V 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23		df-rex 3066	. . . . 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 4995	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
26		eliun 4995	. . . 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 2725	. 2 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
29	1, 28	eqtrid 2779	1 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ∃wrex 3065  Vcvv 3469   ∩ cin 3943   ⊆ wss 3944  {csn 4624  ⟨cop 4630  ∪ ciun 4991   class class class wbr 5142   × cxp 5670  ^◡ccnv 5671  dom cdm 5672  ran crn 5673   “ cima 5675   ∘ ccom 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-iun 4993  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685

This theorem is referenced by:  fparlem3  8113  fparlem4  8114