Theorem domfnex 5870

Description: The domain function is stratified. (Contributed by Scott Fenton, 9-Aug-2019.)

Assertion

Ref	Expression
domfnex	⊢ Dom ∈ V

Proof of Theorem domfnex

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-domfn 5770	. . 3 ⊢ Dom = (x ∈ V ↦ dom x)
2		elin 3219	. . . . . . . . 9 ⊢ (⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) ↔ (⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ Ins4 SI3 Swap ∧ ⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ Ins2 Ins2 S ))
3		vex 2862	. . . . . . . . . . . 12 ⊢ x ∈ V
4	3	oqelins4 5794	. . . . . . . . . . 11 ⊢ (⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ Ins4 SI3 Swap ↔ ⟨{w}, ⟨{z}, {y}⟩⟩ ∈ SI3 Swap )
5		vex 2862	. . . . . . . . . . . . 13 ⊢ w ∈ V
6		vex 2862	. . . . . . . . . . . . 13 ⊢ z ∈ V
7		vex 2862	. . . . . . . . . . . . 13 ⊢ y ∈ V
8	5, 6, 7	otsnelsi3 5805	. . . . . . . . . . . 12 ⊢ (⟨{w}, ⟨{z}, {y}⟩⟩ ∈ SI3 Swap ↔ ⟨w, ⟨z, y⟩⟩ ∈ Swap )
9		df-br 4640	. . . . . . . . . . . 12 ⊢ (w Swap ⟨z, y⟩ ↔ ⟨w, ⟨z, y⟩⟩ ∈ Swap )
10	6, 7	brswap2 4860	. . . . . . . . . . . 12 ⊢ (w Swap ⟨z, y⟩ ↔ w = ⟨y, z⟩)
11	8, 9, 10	3bitr2i 264	. . . . . . . . . . 11 ⊢ (⟨{w}, ⟨{z}, {y}⟩⟩ ∈ SI3 Swap ↔ w = ⟨y, z⟩)
12	4, 11	bitri 240	. . . . . . . . . 10 ⊢ (⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ Ins4 SI3 Swap ↔ w = ⟨y, z⟩)
13		snex 4111	. . . . . . . . . . . 12 ⊢ {z} ∈ V
14	13	otelins2 5791	. . . . . . . . . . 11 ⊢ (⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ Ins2 Ins2 S ↔ ⟨{w}, ⟨{y}, x⟩⟩ ∈ Ins2 S )
15		snex 4111	. . . . . . . . . . . . 13 ⊢ {y} ∈ V
16	15	otelins2 5791	. . . . . . . . . . . 12 ⊢ (⟨{w}, ⟨{y}, x⟩⟩ ∈ Ins2 S ↔ ⟨{w}, x⟩ ∈ S )
17	5, 3	opelssetsn 4760	. . . . . . . . . . . 12 ⊢ (⟨{w}, x⟩ ∈ S ↔ w ∈ x)
18	16, 17	bitri 240	. . . . . . . . . . 11 ⊢ (⟨{w}, ⟨{y}, x⟩⟩ ∈ Ins2 S ↔ w ∈ x)
19	14, 18	bitri 240	. . . . . . . . . 10 ⊢ (⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ Ins2 Ins2 S ↔ w ∈ x)
20	12, 19	anbi12i 678	. . . . . . . . 9 ⊢ ((⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ Ins4 SI3 Swap ∧ ⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ Ins2 Ins2 S ) ↔ (w = ⟨y, z⟩ ∧ w ∈ x))
21	2, 20	bitri 240	. . . . . . . 8 ⊢ (⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) ↔ (w = ⟨y, z⟩ ∧ w ∈ x))
22	21	exbii 1582	. . . . . . 7 ⊢ (∃w⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) ↔ ∃w(w = ⟨y, z⟩ ∧ w ∈ x))
23		elima1c 4947	. . . . . . 7 ⊢ (⟨{z}, ⟨{y}, x⟩⟩ ∈ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ↔ ∃w⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ))
24		df-clel 2349	. . . . . . 7 ⊢ (⟨y, z⟩ ∈ x ↔ ∃w(w = ⟨y, z⟩ ∧ w ∈ x))
25	22, 23, 24	3bitr4i 268	. . . . . 6 ⊢ (⟨{z}, ⟨{y}, x⟩⟩ ∈ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ↔ ⟨y, z⟩ ∈ x)
26	25	exbii 1582	. . . . 5 ⊢ (∃z⟨{z}, ⟨{y}, x⟩⟩ ∈ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ↔ ∃z⟨y, z⟩ ∈ x)
27		elima1c 4947	. . . . 5 ⊢ (⟨{y}, x⟩ ∈ ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) “ 1c) ↔ ∃z⟨{z}, ⟨{y}, x⟩⟩ ∈ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c))
28		eldm2 4899	. . . . 5 ⊢ (y ∈ dom x ↔ ∃z⟨y, z⟩ ∈ x)
29	26, 27, 28	3bitr4i 268	. . . 4 ⊢ (⟨{y}, x⟩ ∈ ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) “ 1c) ↔ y ∈ dom x)
30	29	releqmpt 5808	. . 3 ⊢ ((V × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) “ 1c)) “ 1c)) = (x ∈ V ↦ dom x)
31	1, 30	eqtr4i 2376	. 2 ⊢ Dom = ((V × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) “ 1c)) “ 1c))
32		vvex 4109	. . 3 ⊢ V ∈ V
33		swapex 4742	. . . . . . . 8 ⊢ Swap ∈ V
34	33	si3ex 5806	. . . . . . 7 ⊢ SI3 Swap ∈ V
35	34	ins4ex 5799	. . . . . 6 ⊢ Ins4 SI3 Swap ∈ V
36		ssetex 4744	. . . . . . . 8 ⊢ S ∈ V
37	36	ins2ex 5797	. . . . . . 7 ⊢ Ins2 S ∈ V
38	37	ins2ex 5797	. . . . . 6 ⊢ Ins2 Ins2 S ∈ V
39	35, 38	inex 4105	. . . . 5 ⊢ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) ∈ V
40		1cex 4142	. . . . 5 ⊢ 1c ∈ V
41	39, 40	imaex 4747	. . . 4 ⊢ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∈ V
42	41, 40	imaex 4747	. . 3 ⊢ ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) “ 1c) ∈ V
43	32, 42	mptexlem 5810	. 2 ⊢ ((V × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) “ 1c)) “ 1c)) ∈ V
44	31, 43	eqeltri 2423	1 ⊢ Dom ∈ V

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∼ ccompl 3205   ∩ cin 3208   ⊕ csymdif 3209  {csn 3737  1_cc1c 4134  ⟨cop 4561   class class class wbr 4639   Swap cswap 4718   S csset 4719   “ cima 4722   × cxp 4770  ^◡ccnv 4771  dom cdm 4772   ↦ cmpt 5651   Ins2 cins2 5749   Ins3 cins3 5751   Ins4 cins4 5755   SI₃ csi3 5757   Dom cdomfn 5769

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-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-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-2nd 4797  df-mpt 5652  df-txp 5736  df-ins2 5750  df-ins3 5752  df-ins4 5756  df-si3 5758  df-domfn 5770

This theorem is referenced by: (None)