Theorem releqmpt2 5810

Description: Equality condition for a mapping operation. (Contributed by SF, 13-Feb-2015.)

Hypothesis

Ref	Expression
releqmpt2.1	⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ R ↔ z ∈ V)

Assertion

Ref	Expression
releqmpt2	⊢ (((A × B) × V) ∖ (( Ins2 S ⊕ Ins3 R) “ 1c)) = (x ∈ A, y ∈ B ↦ V)

Distinct variable groups:   x,A,y   x,B,y   x,R,y,z   z,V   x,y,z

Allowed substitution hints:   A(z)   B(z)   V(x,y)

Proof of Theorem releqmpt2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3222	. . . 4 ⊢ (⟨⟨x, y⟩, w⟩ ∈ (((A × B) × V) ∖ (( Ins2 S ⊕ Ins3 R) “ 1c)) ↔ (⟨⟨x, y⟩, w⟩ ∈ ((A × B) × V) ∧ ¬ ⟨⟨x, y⟩, w⟩ ∈ (( Ins2 S ⊕ Ins3 R) “ 1c)))
2		vex 2863	. . . . . . 7 ⊢ w ∈ V
3		opelxp 4812	. . . . . . 7 ⊢ (⟨⟨x, y⟩, w⟩ ∈ ((A × B) × V) ↔ (⟨x, y⟩ ∈ (A × B) ∧ w ∈ V))
4	2, 3	mpbiran2 885	. . . . . 6 ⊢ (⟨⟨x, y⟩, w⟩ ∈ ((A × B) × V) ↔ ⟨x, y⟩ ∈ (A × B))
5		opelxp 4812	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (A × B) ↔ (x ∈ A ∧ y ∈ B))
6	4, 5	bitri 240	. . . . 5 ⊢ (⟨⟨x, y⟩, w⟩ ∈ ((A × B) × V) ↔ (x ∈ A ∧ y ∈ B))
7		dfcleq 2347	. . . . . 6 ⊢ (w = V ↔ ∀z(z ∈ w ↔ z ∈ V))
8		elima1c 4948	. . . . . . . 8 ⊢ (⟨⟨x, y⟩, w⟩ ∈ (( Ins2 S ⊕ Ins3 R) “ 1c) ↔ ∃z⟨{z}, ⟨⟨x, y⟩, w⟩⟩ ∈ ( Ins2 S ⊕ Ins3 R))
9		elsymdif 3224	. . . . . . . . . 10 ⊢ (⟨{z}, ⟨⟨x, y⟩, w⟩⟩ ∈ ( Ins2 S ⊕ Ins3 R) ↔ ¬ (⟨{z}, ⟨⟨x, y⟩, w⟩⟩ ∈ Ins2 S ↔ ⟨{z}, ⟨⟨x, y⟩, w⟩⟩ ∈ Ins3 R))
10		vex 2863	. . . . . . . . . . . . . 14 ⊢ x ∈ V
11		vex 2863	. . . . . . . . . . . . . 14 ⊢ y ∈ V
12	10, 11	opex 4589	. . . . . . . . . . . . 13 ⊢ ⟨x, y⟩ ∈ V
13	12	otelins2 5792	. . . . . . . . . . . 12 ⊢ (⟨{z}, ⟨⟨x, y⟩, w⟩⟩ ∈ Ins2 S ↔ ⟨{z}, w⟩ ∈ S )
14		vex 2863	. . . . . . . . . . . . 13 ⊢ z ∈ V
15	14, 2	opelssetsn 4761	. . . . . . . . . . . 12 ⊢ (⟨{z}, w⟩ ∈ S ↔ z ∈ w)
16	13, 15	bitri 240	. . . . . . . . . . 11 ⊢ (⟨{z}, ⟨⟨x, y⟩, w⟩⟩ ∈ Ins2 S ↔ z ∈ w)
17	2	otelins3 5793	. . . . . . . . . . . 12 ⊢ (⟨{z}, ⟨⟨x, y⟩, w⟩⟩ ∈ Ins3 R ↔ ⟨{z}, ⟨x, y⟩⟩ ∈ R)
18		releqmpt2.1	. . . . . . . . . . . 12 ⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ R ↔ z ∈ V)
19	17, 18	bitri 240	. . . . . . . . . . 11 ⊢ (⟨{z}, ⟨⟨x, y⟩, w⟩⟩ ∈ Ins3 R ↔ z ∈ V)
20	16, 19	bibi12i 306	. . . . . . . . . 10 ⊢ ((⟨{z}, ⟨⟨x, y⟩, w⟩⟩ ∈ Ins2 S ↔ ⟨{z}, ⟨⟨x, y⟩, w⟩⟩ ∈ Ins3 R) ↔ (z ∈ w ↔ z ∈ V))
21	9, 20	xchbinx 301	. . . . . . . . 9 ⊢ (⟨{z}, ⟨⟨x, y⟩, w⟩⟩ ∈ ( Ins2 S ⊕ Ins3 R) ↔ ¬ (z ∈ w ↔ z ∈ V))
22	21	exbii 1582	. . . . . . . 8 ⊢ (∃z⟨{z}, ⟨⟨x, y⟩, w⟩⟩ ∈ ( Ins2 S ⊕ Ins3 R) ↔ ∃z ¬ (z ∈ w ↔ z ∈ V))
23		exnal 1574	. . . . . . . 8 ⊢ (∃z ¬ (z ∈ w ↔ z ∈ V) ↔ ¬ ∀z(z ∈ w ↔ z ∈ V))
24	8, 22, 23	3bitri 262	. . . . . . 7 ⊢ (⟨⟨x, y⟩, w⟩ ∈ (( Ins2 S ⊕ Ins3 R) “ 1c) ↔ ¬ ∀z(z ∈ w ↔ z ∈ V))
25	24	con2bii 322	. . . . . 6 ⊢ (∀z(z ∈ w ↔ z ∈ V) ↔ ¬ ⟨⟨x, y⟩, w⟩ ∈ (( Ins2 S ⊕ Ins3 R) “ 1c))
26	7, 25	bitr2i 241	. . . . 5 ⊢ (¬ ⟨⟨x, y⟩, w⟩ ∈ (( Ins2 S ⊕ Ins3 R) “ 1c) ↔ w = V)
27	6, 26	anbi12i 678	. . . 4 ⊢ ((⟨⟨x, y⟩, w⟩ ∈ ((A × B) × V) ∧ ¬ ⟨⟨x, y⟩, w⟩ ∈ (( Ins2 S ⊕ Ins3 R) “ 1c)) ↔ ((x ∈ A ∧ y ∈ B) ∧ w = V))
28	1, 27	bitri 240	. . 3 ⊢ (⟨⟨x, y⟩, w⟩ ∈ (((A × B) × V) ∖ (( Ins2 S ⊕ Ins3 R) “ 1c)) ↔ ((x ∈ A ∧ y ∈ B) ∧ w = V))
29	28	oprabbi2i 5648	. 2 ⊢ (((A × B) × V) ∖ (( Ins2 S ⊕ Ins3 R) “ 1c)) = {⟨⟨x, y⟩, w⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ w = V)}
30		df-mpt2 5655	. 2 ⊢ (x ∈ A, y ∈ B ↦ V) = {⟨⟨x, y⟩, w⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ w = V)}
31	29, 30	eqtr4i 2376	1 ⊢ (((A × B) × V) ∖ (( Ins2 S ⊕ Ins3 R) “ 1c)) = (x ∈ A, y ∈ B ↦ V)

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∖ cdif 3207   ⊕ csymdif 3210  {csn 3738  1_cc1c 4135  ⟨cop 4562   S csset 4720   “ cima 4723   × cxp 4771  {coprab 5528   ↦ cmpt2 5654   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-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-ins2 5751  df-ins3 5753

This theorem is referenced by:  cupex  5817  composeex  5821  addcfnex  5825  crossex  5851  mucex  6134  ceex  6175