Theorem releqmpt 5808

Description: Equality condition for a mapping. (Contributed by SF, 9-Mar-2015.)

Hypothesis

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

Assertion

Ref	Expression
releqmpt	⊢ ((A × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c)) = (x ∈ A ↦ V)

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

Allowed substitution hints:   A(y)   V(x)

Proof of Theorem releqmpt

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3219	. . . 4 ⊢ (⟨x, z⟩ ∈ ((A × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c)) ↔ (⟨x, z⟩ ∈ (A × V) ∧ ⟨x, z⟩ ∈ ◡ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c)))
2		vex 2862	. . . . . 6 ⊢ z ∈ V
3		opelxp 4811	. . . . . 6 ⊢ (⟨x, z⟩ ∈ (A × V) ↔ (x ∈ A ∧ z ∈ V))
4	2, 3	mpbiran2 885	. . . . 5 ⊢ (⟨x, z⟩ ∈ (A × V) ↔ x ∈ A)
5		opelcnv 4893	. . . . . 6 ⊢ (⟨x, z⟩ ∈ ◡ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c) ↔ ⟨z, x⟩ ∈ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c))
6		vex 2862	. . . . . . 7 ⊢ x ∈ V
7		releqmpt.1	. . . . . . 7 ⊢ (⟨{y}, x⟩ ∈ R ↔ y ∈ V)
8	6, 7	releqel 5807	. . . . . 6 ⊢ (⟨z, x⟩ ∈ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c) ↔ z = V)
9	5, 8	bitri 240	. . . . 5 ⊢ (⟨x, z⟩ ∈ ◡ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c) ↔ z = V)
10	4, 9	anbi12i 678	. . . 4 ⊢ ((⟨x, z⟩ ∈ (A × V) ∧ ⟨x, z⟩ ∈ ◡ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c)) ↔ (x ∈ A ∧ z = V))
11	1, 10	bitri 240	. . 3 ⊢ (⟨x, z⟩ ∈ ((A × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c)) ↔ (x ∈ A ∧ z = V))
12	11	opabbi2i 4866	. 2 ⊢ ((A × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c)) = {⟨x, z⟩ ∣ (x ∈ A ∧ z = V)}
13		df-mpt 5652	. 2 ⊢ (x ∈ A ↦ V) = {⟨x, z⟩ ∣ (x ∈ A ∧ z = V)}
14	12, 13	eqtr4i 2376	1 ⊢ ((A × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c)) = (x ∈ A ↦ V)

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∼ ccompl 3205   ∩ cin 3208   ⊕ csymdif 3209  {csn 3737  1_cc1c 4134  ⟨cop 4561  {copab 4622   S csset 4719   “ cima 4722   × cxp 4770  ^◡ccnv 4771   ↦ cmpt 5651   Ins2 cins2 5749   Ins3 cins3 5751

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-sset 4725  df-co 4726  df-ima 4727  df-xp 4784  df-cnv 4785  df-2nd 4797  df-mpt 5652  df-txp 5736  df-ins2 5750  df-ins3 5752

This theorem is referenced by:  pw1fnex  5852  domfnex  5870  ranfnex  5871  enprmaplem1  6076  enprmaplem4  6079  tcfnex  6244