Theorem releqmpt 5809

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 3220	. . . 4 ⊢ (⟨x, z⟩ ∈ ((A × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c)) ↔ (⟨x, z⟩ ∈ (A × V) ∧ ⟨x, z⟩ ∈ ◡ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c)))
2		vex 2863	. . . . . 6 ⊢ z ∈ V
3		opelxp 4812	. . . . . 6 ⊢ (⟨x, z⟩ ∈ (A × V) ↔ (x ∈ A ∧ z ∈ V))
4	2, 3	mpbiran2 885	. . . . 5 ⊢ (⟨x, z⟩ ∈ (A × V) ↔ x ∈ A)
5		opelcnv 4894	. . . . . 6 ⊢ (⟨x, z⟩ ∈ ◡ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c) ↔ ⟨z, x⟩ ∈ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c))
6		vex 2863	. . . . . . 7 ⊢ x ∈ V
7		releqmpt.1	. . . . . . 7 ⊢ (⟨{y}, x⟩ ∈ R ↔ y ∈ V)
8	6, 7	releqel 5808	. . . . . 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 4867	. 2 ⊢ ((A × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c)) = {⟨x, z⟩ ∣ (x ∈ A ∧ z = V)}
13		df-mpt 5653	. 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 2860   ∼ ccompl 3206   ∩ cin 3209   ⊕ csymdif 3210  {csn 3738  1_cc1c 4135  ⟨cop 4562  {copab 4623   S csset 4720   “ cima 4723   × cxp 4771  ^◡ccnv 4772   ↦ cmpt 5652   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-mpt 5653  df-txp 5737  df-ins2 5751  df-ins3 5753

This theorem is referenced by:  pw1fnex  5853  domfnex  5871  ranfnex  5872  enprmaplem1  6077  enprmaplem4  6080  tcfnex  6245