Theorem domfnex 5871

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 5771	. . 3 ⊢ Dom = (x ∈ V ↦ dom x)
2		elin 3220	. . . . . . . . 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 2863	. . . . . . . . . . . 12 ⊢ x ∈ V
4	3	oqelins4 5795	. . . . . . . . . . 11 ⊢ (⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ Ins4 SI3 Swap ↔ ⟨{w}, ⟨{z}, {y}⟩⟩ ∈ SI3 Swap )
5		vex 2863	. . . . . . . . . . . . 13 ⊢ w ∈ V
6		vex 2863	. . . . . . . . . . . . 13 ⊢ z ∈ V
7		vex 2863	. . . . . . . . . . . . 13 ⊢ y ∈ V
8	5, 6, 7	otsnelsi3 5806	. . . . . . . . . . . 12 ⊢ (⟨{w}, ⟨{z}, {y}⟩⟩ ∈ SI3 Swap ↔ ⟨w, ⟨z, y⟩⟩ ∈ Swap )
9		df-br 4641	. . . . . . . . . . . 12 ⊢ (w Swap ⟨z, y⟩ ↔ ⟨w, ⟨z, y⟩⟩ ∈ Swap )
10	6, 7	brswap2 4861	. . . . . . . . . . . 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 4112	. . . . . . . . . . . 12 ⊢ {z} ∈ V
14	13	otelins2 5792	. . . . . . . . . . 11 ⊢ (⟨{w}, ⟨{z}, ⟨{y}, x⟩⟩⟩ ∈ Ins2 Ins2 S ↔ ⟨{w}, ⟨{y}, x⟩⟩ ∈ Ins2 S )
15		snex 4112	. . . . . . . . . . . . 13 ⊢ {y} ∈ V
16	15	otelins2 5792	. . . . . . . . . . . 12 ⊢ (⟨{w}, ⟨{y}, x⟩⟩ ∈ Ins2 S ↔ ⟨{w}, x⟩ ∈ S )
17	5, 3	opelssetsn 4761	. . . . . . . . . . . 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 4948	. . . . . . 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 4948	. . . . 5 ⊢ (⟨{y}, x⟩ ∈ ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) “ 1c) ↔ ∃z⟨{z}, ⟨{y}, x⟩⟩ ∈ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c))
28		eldm2 4900	. . . . 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 5809	. . 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 4110	. . 3 ⊢ V ∈ V
33		swapex 4743	. . . . . . . 8 ⊢ Swap ∈ V
34	33	si3ex 5807	. . . . . . 7 ⊢ SI3 Swap ∈ V
35	34	ins4ex 5800	. . . . . 6 ⊢ Ins4 SI3 Swap ∈ V
36		ssetex 4745	. . . . . . . 8 ⊢ S ∈ V
37	36	ins2ex 5798	. . . . . . 7 ⊢ Ins2 S ∈ V
38	37	ins2ex 5798	. . . . . 6 ⊢ Ins2 Ins2 S ∈ V
39	35, 38	inex 4106	. . . . 5 ⊢ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) ∈ V
40		1cex 4143	. . . . 5 ⊢ 1c ∈ V
41	39, 40	imaex 4748	. . . 4 ⊢ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∈ V
42	41, 40	imaex 4748	. . 3 ⊢ ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) “ 1c) ∈ V
43	32, 42	mptexlem 5811	. 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 2860   ∼ ccompl 3206   ∩ cin 3209   ⊕ csymdif 3210  {csn 3738  1_cc1c 4135  ⟨cop 4562   class class class wbr 4640   Swap cswap 4719   S csset 4720   “ cima 4723   × cxp 4771  ^◡ccnv 4772  dom cdm 4773   ↦ cmpt 5652   Ins2 cins2 5750   Ins3 cins3 5752   Ins4 cins4 5756   SI₃ csi3 5758   Dom cdomfn 5770

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-2nd 4798  df-mpt 5653  df-txp 5737  df-ins2 5751  df-ins3 5753  df-ins4 5757  df-si3 5759  df-domfn 5771

This theorem is referenced by: (None)