Theorem dfco2a 5081

Description: Generalization of dfco2 5080, where C can have any value between dom A ∩ ran B and V. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 21-Dec-2008.) (Revised by set.mm contributors, 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 5080	. 2 ⊢ (A ∘ B) = ∪x ∈ V ((◡B “ {x}) × (A “ {x}))
2		elimasn 5019	. . . . . . . . . . . . 13 ⊢ (w ∈ (A “ {x}) ↔ ⟨x, w⟩ ∈ A)
3		opeldm 4910	. . . . . . . . . . . . 13 ⊢ (⟨x, w⟩ ∈ A → x ∈ dom A)
4	2, 3	sylbi 187	. . . . . . . . . . . 12 ⊢ (w ∈ (A “ {x}) → x ∈ dom A)
5		eliniseg 5020	. . . . . . . . . . . . 13 ⊢ (z ∈ (◡B “ {x}) ↔ zBx)
6		brelrn 4960	. . . . . . . . . . . . 13 ⊢ (zBx → x ∈ ran B)
7	5, 6	sylbi 187	. . . . . . . . . . . 12 ⊢ (z ∈ (◡B “ {x}) → x ∈ ran B)
8	4, 7	anim12i 549	. . . . . . . . . . 11 ⊢ ((w ∈ (A “ {x}) ∧ z ∈ (◡B “ {x})) → (x ∈ dom A ∧ x ∈ ran B))
9	8	ancoms 439	. . . . . . . . . 10 ⊢ ((z ∈ (◡B “ {x}) ∧ w ∈ (A “ {x})) → (x ∈ dom A ∧ x ∈ ran B))
10	9	adantl 452	. . . . . . . . 9 ⊢ ((y = ⟨z, w⟩ ∧ (z ∈ (◡B “ {x}) ∧ w ∈ (A “ {x}))) → (x ∈ dom A ∧ x ∈ ran B))
11	10	exlimivv 1635	. . . . . . . 8 ⊢ (∃z∃w(y = ⟨z, w⟩ ∧ (z ∈ (◡B “ {x}) ∧ w ∈ (A “ {x}))) → (x ∈ dom A ∧ x ∈ ran B))
12		elxp 4801	. . . . . . . 8 ⊢ (y ∈ ((◡B “ {x}) × (A “ {x})) ↔ ∃z∃w(y = ⟨z, w⟩ ∧ (z ∈ (◡B “ {x}) ∧ w ∈ (A “ {x}))))
13		elin 3219	. . . . . . . 8 ⊢ (x ∈ (dom A ∩ ran B) ↔ (x ∈ dom A ∧ x ∈ ran B))
14	11, 12, 13	3imtr4i 257	. . . . . . 7 ⊢ (y ∈ ((◡B “ {x}) × (A “ {x})) → x ∈ (dom A ∩ ran B))
15		ssel 3267	. . . . . . 7 ⊢ ((dom A ∩ ran B) ⊆ C → (x ∈ (dom A ∩ ran B) → x ∈ C))
16	14, 15	syl5 28	. . . . . 6 ⊢ ((dom A ∩ ran B) ⊆ C → (y ∈ ((◡B “ {x}) × (A “ {x})) → x ∈ C))
17	16	pm4.71rd 616	. . . . 5 ⊢ ((dom A ∩ ran B) ⊆ C → (y ∈ ((◡B “ {x}) × (A “ {x})) ↔ (x ∈ C ∧ y ∈ ((◡B “ {x}) × (A “ {x})))))
18	17	exbidv 1626	. . . 4 ⊢ ((dom A ∩ ran B) ⊆ C → (∃x y ∈ ((◡B “ {x}) × (A “ {x})) ↔ ∃x(x ∈ C ∧ y ∈ ((◡B “ {x}) × (A “ {x})))))
19		eliun 3973	. . . . 5 ⊢ (y ∈ ∪x ∈ V ((◡B “ {x}) × (A “ {x})) ↔ ∃x ∈ V y ∈ ((◡B “ {x}) × (A “ {x})))
20		rexv 2873	. . . . 5 ⊢ (∃x ∈ V y ∈ ((◡B “ {x}) × (A “ {x})) ↔ ∃x y ∈ ((◡B “ {x}) × (A “ {x})))
21	19, 20	bitri 240	. . . 4 ⊢ (y ∈ ∪x ∈ V ((◡B “ {x}) × (A “ {x})) ↔ ∃x y ∈ ((◡B “ {x}) × (A “ {x})))
22		eliun 3973	. . . . 5 ⊢ (y ∈ ∪x ∈ C ((◡B “ {x}) × (A “ {x})) ↔ ∃x ∈ C y ∈ ((◡B “ {x}) × (A “ {x})))
23		df-rex 2620	. . . . 5 ⊢ (∃x ∈ C y ∈ ((◡B “ {x}) × (A “ {x})) ↔ ∃x(x ∈ C ∧ y ∈ ((◡B “ {x}) × (A “ {x}))))
24	22, 23	bitri 240	. . . 4 ⊢ (y ∈ ∪x ∈ C ((◡B “ {x}) × (A “ {x})) ↔ ∃x(x ∈ C ∧ y ∈ ((◡B “ {x}) × (A “ {x}))))
25	18, 21, 24	3bitr4g 279	. . 3 ⊢ ((dom A ∩ ran B) ⊆ C → (y ∈ ∪x ∈ V ((◡B “ {x}) × (A “ {x})) ↔ y ∈ ∪x ∈ C ((◡B “ {x}) × (A “ {x}))))
26	25	eqrdv 2351	. 2 ⊢ ((dom A ∩ ran B) ⊆ C → ∪x ∈ V ((◡B “ {x}) × (A “ {x})) = ∪x ∈ C ((◡B “ {x}) × (A “ {x})))
27	1, 26	syl5eq 2397	1 ⊢ ((dom A ∩ ran B) ⊆ C → (A ∘ B) = ∪x ∈ C ((◡B “ {x}) × (A “ {x})))

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859   ∩ cin 3208   ⊆ wss 3257  {csn 3737  ∪ciun 3969  ⟨cop 4561   class class class wbr 4639   ∘ ccom 4721   “ cima 4722   × cxp 4770  ^◡ccnv 4771  dom cdm 4772  ran crn 4773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788

This theorem is referenced by: (None)