Theorem dfco2a 4808

Description: Generalization of dfco2 4807, 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 4807	. 2 ⊢ (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
2		vex 2557	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
3		vex 2557	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
4	3	eliniseg 4682	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ V → (𝑧 ∈ (◡𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥))
5	2, 4	ax-mp 7	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (◡𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥)
6	3, 2	brelrn 4554	. . . . . . . . . . . . 13 ⊢ (𝑧𝐵𝑥 → 𝑥 ∈ ran 𝐵)
7	5, 6	sylbi 114	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (◡𝐵 “ {𝑥}) → 𝑥 ∈ ran 𝐵)
8		vex 2557	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
9	2, 8	elimasn 4679	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐴)
10	2, 8	opeldm 4525	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑤⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
11	9, 10	sylbi 114	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (𝐴 “ {𝑥}) → 𝑥 ∈ dom 𝐴)
12	7, 11	anim12ci 322	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥})) → (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
13	12	adantl 262	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
14	13	exlimivv 1776	. . . . . . . . 9 ⊢ (∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
15		elxp 4349	. . . . . . . . 9 ⊢ (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))))
16		elin 3123	. . . . . . . . 9 ⊢ (𝑥 ∈ (dom 𝐴 ∩ ran 𝐵) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
17	14, 15, 16	3imtr4i 190	. . . . . . . 8 ⊢ (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) → 𝑥 ∈ (dom 𝐴 ∩ ran 𝐵))
18		ssel 2936	. . . . . . . 8 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑥 ∈ (dom 𝐴 ∩ ran 𝐵) → 𝑥 ∈ 𝐶))
19	17, 18	syl5 28	. . . . . . 7 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) → 𝑥 ∈ 𝐶))
20	19	pm4.71rd 374	. . . . . 6 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
21	20	exbidv 1706	. . . . 5 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
22		rexv 2569	. . . . 5 ⊢ (∃𝑥 ∈ V 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23		df-rex 2309	. . . . 5 ⊢ (∃𝑥 ∈ 𝐶 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
24	21, 22, 23	3bitr4g 212	. . . 4 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 ∈ V 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ 𝐶 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
25		eliun 3658	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
26		eliun 3658	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ 𝐶 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
27	24, 25, 26	3bitr4g 212	. . 3 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
28	27	eqrdv 2038	. 2 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
29	1, 28	syl5eq 2084	1 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2304  Vcvv 2554   ∩ cin 2913   ⊆ wss 2914  {csn 3372  ⟨cop 3375  ∪ ciun 3654   class class class wbr 3761   × cxp 4330  ^◡ccnv 4331  dom cdm 4332  ran crn 4333   “ cima 4335   ∘ ccom 4336

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-iun 3656  df-br 3762  df-opab 3816  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345

This theorem is referenced by: (None)