Theorem dfco2a 4748

Description: Generalization of dfco2 4747, 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 4747	. 2 ⊢ (A ∘ B) = ∪ x ∈ V ((◡B “ {x}) × (A “ {x}))
2		vex 2538	. . . . . . . . . . . . . 14 ⊢ x ∈ V
3		vex 2538	. . . . . . . . . . . . . . 15 ⊢ z ∈ V
4	3	eliniseg 4622	. . . . . . . . . . . . . 14 ⊢ (x ∈ V → (z ∈ (◡B “ {x}) ↔ zBx))
5	2, 4	ax-mp 7	. . . . . . . . . . . . 13 ⊢ (z ∈ (◡B “ {x}) ↔ zBx)
6	3, 2	brelrn 4494	. . . . . . . . . . . . 13 ⊢ (zBx → x ∈ ran B)
7	5, 6	sylbi 114	. . . . . . . . . . . 12 ⊢ (z ∈ (◡B “ {x}) → x ∈ ran B)
8		vex 2538	. . . . . . . . . . . . . 14 ⊢ w ∈ V
9	2, 8	elimasn 4619	. . . . . . . . . . . . 13 ⊢ (w ∈ (A “ {x}) ↔ ⟨x, w⟩ ∈ A)
10	2, 8	opeldm 4465	. . . . . . . . . . . . 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 1758	. . . . . . . . 9 ⊢ (∃z∃w(y = ⟨z, w⟩ ∧ (z ∈ (◡B “ {x}) ∧ w ∈ (A “ {x}))) → (x ∈ dom A ∧ x ∈ ran B))
15		elxp 4289	. . . . . . . . 9 ⊢ (y ∈ ((◡B “ {x}) × (A “ {x})) ↔ ∃z∃w(y = ⟨z, w⟩ ∧ (z ∈ (◡B “ {x}) ∧ w ∈ (A “ {x}))))
16		elin 3103	. . . . . . . . 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 2916	. . . . . . . 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 1688	. . . . 5 ⊢ ((dom A ∩ ran B) ⊆ 𝐶 → (∃x y ∈ ((◡B “ {x}) × (A “ {x})) ↔ ∃x(x ∈ 𝐶 ∧ y ∈ ((◡B “ {x}) × (A “ {x})))))
22		rexv 2549	. . . . 5 ⊢ (∃x ∈ V y ∈ ((◡B “ {x}) × (A “ {x})) ↔ ∃x y ∈ ((◡B “ {x}) × (A “ {x})))
23		df-rex 2290	. . . . 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 3635	. . . 4 ⊢ (y ∈ ∪ x ∈ V ((◡B “ {x}) × (A “ {x})) ↔ ∃x ∈ V y ∈ ((◡B “ {x}) × (A “ {x})))
26		eliun 3635	. . . 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 2020	. 2 ⊢ ((dom A ∩ ran B) ⊆ 𝐶 → ∪ x ∈ V ((◡B “ {x}) × (A “ {x})) = ∪ x ∈ 𝐶 ((◡B “ {x}) × (A “ {x})))
29	1, 28	syl5eq 2066	1 ⊢ ((dom A ∩ ran B) ⊆ 𝐶 → (A ∘ B) = ∪ x ∈ 𝐶 ((◡B “ {x}) × (A “ {x})))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∃wrex 2285  Vcvv 2535   ∩ cin 2893   ⊆ wss 2894  {csn 3350  ⟨cop 3353  ∪ ciun 3631   class class class wbr 3738   × cxp 4270  ^◡ccnv 4271  dom cdm 4272  ran crn 4273   “ cima 4275   ∘ ccom 4276

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-iun 3633  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285

This theorem is referenced by: (None)