Theorem cupex 5817

Description: The little cup function is a set. (Contributed by SF, 11-Feb-2015.)

Assertion

Ref	Expression
cupex	⊢ Cup ∈ V

Proof of Theorem cupex

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

Step	Hyp	Ref	Expression
1		df-cup 5743	. . 3 ⊢ Cup = (x ∈ V, y ∈ V ↦ (x ∪ y))
2		vex 2863	. . . . . . . 8 ⊢ y ∈ V
3	2	otelins3 5793	. . . . . . 7 ⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ Ins3 S ↔ ⟨{z}, x⟩ ∈ S )
4		vex 2863	. . . . . . . 8 ⊢ z ∈ V
5		vex 2863	. . . . . . . 8 ⊢ x ∈ V
6	4, 5	opelssetsn 4761	. . . . . . 7 ⊢ (⟨{z}, x⟩ ∈ S ↔ z ∈ x)
7	3, 6	bitri 240	. . . . . 6 ⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ Ins3 S ↔ z ∈ x)
8	5	otelins2 5792	. . . . . . 7 ⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ Ins2 S ↔ ⟨{z}, y⟩ ∈ S )
9	4, 2	opelssetsn 4761	. . . . . . 7 ⊢ (⟨{z}, y⟩ ∈ S ↔ z ∈ y)
10	8, 9	bitri 240	. . . . . 6 ⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ Ins2 S ↔ z ∈ y)
11	7, 10	orbi12i 507	. . . . 5 ⊢ ((⟨{z}, ⟨x, y⟩⟩ ∈ Ins3 S ∨ ⟨{z}, ⟨x, y⟩⟩ ∈ Ins2 S ) ↔ (z ∈ x ∨ z ∈ y))
12		elun 3221	. . . . 5 ⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ ( Ins3 S ∪ Ins2 S ) ↔ (⟨{z}, ⟨x, y⟩⟩ ∈ Ins3 S ∨ ⟨{z}, ⟨x, y⟩⟩ ∈ Ins2 S ))
13		elun 3221	. . . . 5 ⊢ (z ∈ (x ∪ y) ↔ (z ∈ x ∨ z ∈ y))
14	11, 12, 13	3bitr4i 268	. . . 4 ⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ ( Ins3 S ∪ Ins2 S ) ↔ z ∈ (x ∪ y))
15	14	releqmpt2 5810	. . 3 ⊢ (((V × V) × V) ∖ (( Ins2 S ⊕ Ins3 ( Ins3 S ∪ Ins2 S )) “ 1c)) = (x ∈ V, y ∈ V ↦ (x ∪ y))
16	1, 15	eqtr4i 2376	. 2 ⊢ Cup = (((V × V) × V) ∖ (( Ins2 S ⊕ Ins3 ( Ins3 S ∪ Ins2 S )) “ 1c))
17		vvex 4110	. . 3 ⊢ V ∈ V
18		ssetex 4745	. . . . 5 ⊢ S ∈ V
19	18	ins3ex 5799	. . . 4 ⊢ Ins3 S ∈ V
20	18	ins2ex 5798	. . . 4 ⊢ Ins2 S ∈ V
21	19, 20	unex 4107	. . 3 ⊢ ( Ins3 S ∪ Ins2 S ) ∈ V
22	17, 17, 21	mpt2exlem 5812	. 2 ⊢ (((V × V) × V) ∖ (( Ins2 S ⊕ Ins3 ( Ins3 S ∪ Ins2 S )) “ 1c)) ∈ V
23	16, 22	eqeltri 2423	1 ⊢ Cup ∈ V

Syntax hints:   ∨ wo 357   ∈ wcel 1710  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ⊕ csymdif 3210  {csn 3738  1_cc1c 4135  ⟨cop 4562   S csset 4720   “ cima 4723   × cxp 4771   ↦ cmpt2 5654   Cup ccup 5742   Ins2 cins2 5750   Ins3 cins3 5752

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-xp 4785  df-cnv 4786  df-2nd 4798  df-oprab 5529  df-mpt2 5655  df-txp 5737  df-cup 5743  df-ins2 5751  df-ins3 5753

This theorem is referenced by:  addcfnex  5825