Theorem dfco2a 4764

Description: Generalization of dfco2 4763, 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 4763	. 2 ⊢ (A ∘ B) = ∪ x ∈ V ((◡B “ {x}) × (A “ {x}))
2		vex 2554	. . . . . . . . . . . . . 14 ⊢ x ∈ V
3		vex 2554	. . . . . . . . . . . . . . 15 ⊢ z ∈ V
4	3	eliniseg 4638	. . . . . . . . . . . . . 14 ⊢ (x ∈ V → (z ∈ (◡B “ {x}) ↔ zBx))
5	2, 4	ax-mp 7	. . . . . . . . . . . . 13 ⊢ (z ∈ (◡B “ {x}) ↔ zBx)
6	3, 2	brelrn 4510	. . . . . . . . . . . . 13 ⊢ (zBx → x ∈ ran B)
7	5, 6	sylbi 114	. . . . . . . . . . . 12 ⊢ (z ∈ (◡B “ {x}) → x ∈ ran B)
8		vex 2554	. . . . . . . . . . . . . 14 ⊢ w ∈ V
9	2, 8	elimasn 4635	. . . . . . . . . . . . 13 ⊢ (w ∈ (A “ {x}) ↔ ⟨x, w⟩ ∈ A)
10	2, 8	opeldm 4481	. . . . . . . . . . . . 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 1773	. . . . . . . . 9 ⊢ (∃z∃w(y = ⟨z, w⟩ ∧ (z ∈ (◡B “ {x}) ∧ w ∈ (A “ {x}))) → (x ∈ dom A ∧ x ∈ ran B))
15		elxp 4305	. . . . . . . . 9 ⊢ (y ∈ ((◡B “ {x}) × (A “ {x})) ↔ ∃z∃w(y = ⟨z, w⟩ ∧ (z ∈ (◡B “ {x}) ∧ w ∈ (A “ {x}))))
16		elin 3120	. . . . . . . . 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 2933	. . . . . . . 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 1703	. . . . 5 ⊢ ((dom A ∩ ran B) ⊆ 𝐶 → (∃x y ∈ ((◡B “ {x}) × (A “ {x})) ↔ ∃x(x ∈ 𝐶 ∧ y ∈ ((◡B “ {x}) × (A “ {x})))))
22		rexv 2566	. . . . 5 ⊢ (∃x ∈ V y ∈ ((◡B “ {x}) × (A “ {x})) ↔ ∃x y ∈ ((◡B “ {x}) × (A “ {x})))
23		df-rex 2306	. . . . 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 3652	. . . 4 ⊢ (y ∈ ∪ x ∈ V ((◡B “ {x}) × (A “ {x})) ↔ ∃x ∈ V y ∈ ((◡B “ {x}) × (A “ {x})))
26		eliun 3652	. . . 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 2035	. 2 ⊢ ((dom A ∩ ran B) ⊆ 𝐶 → ∪ x ∈ V ((◡B “ {x}) × (A “ {x})) = ∪ x ∈ 𝐶 ((◡B “ {x}) × (A “ {x})))
29	1, 28	syl5eq 2081	1 ⊢ ((dom A ∩ ran B) ⊆ 𝐶 → (A ∘ B) = ∪ x ∈ 𝐶 ((◡B “ {x}) × (A “ {x})))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  Vcvv 2551   ∩ cin 2910   ⊆ wss 2911  {csn 3367  ⟨cop 3370  ∪ ciun 3648   class class class wbr 3755   × cxp 4286  ^◡ccnv 4287  dom cdm 4288  ran crn 4289   “ cima 4291   ∘ ccom 4292

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301

This theorem is referenced by: (None)