Theorem rnsnop 5075

Description: The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by set.mm contributors, 24-Jul-2004.)

Hypothesis

Ref	Expression
rnsnop.1	⊢ A ∈ V

Assertion

Ref	Expression
rnsnop	⊢ ran {⟨A, B⟩} = {B}

Proof of Theorem rnsnop

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

Step	Hyp	Ref	Expression
1		df-br 4640	. . . . . 6 ⊢ (x{⟨A, B⟩}y ↔ ⟨x, y⟩ ∈ {⟨A, B⟩})
2		vex 2862	. . . . . . . . 9 ⊢ x ∈ V
3		vex 2862	. . . . . . . . 9 ⊢ y ∈ V
4	2, 3	opex 4588	. . . . . . . 8 ⊢ ⟨x, y⟩ ∈ V
5	4	elsnc 3756	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
6		opth 4602	. . . . . . 7 ⊢ (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ∧ y = B))
7	5, 6	bitri 240	. . . . . 6 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ (x = A ∧ y = B))
8	1, 7	bitri 240	. . . . 5 ⊢ (x{⟨A, B⟩}y ↔ (x = A ∧ y = B))
9	8	exbii 1582	. . . 4 ⊢ (∃x x{⟨A, B⟩}y ↔ ∃x(x = A ∧ y = B))
10		rnsnop.1	. . . . 5 ⊢ A ∈ V
11		biidd 228	. . . . 5 ⊢ (x = A → (y = B ↔ y = B))
12	10, 11	ceqsexv 2894	. . . 4 ⊢ (∃x(x = A ∧ y = B) ↔ y = B)
13	9, 12	bitri 240	. . 3 ⊢ (∃x x{⟨A, B⟩}y ↔ y = B)
14		elrn 4896	. . 3 ⊢ (y ∈ ran {⟨A, B⟩} ↔ ∃x x{⟨A, B⟩}y)
15	3	elsnc 3756	. . 3 ⊢ (y ∈ {B} ↔ y = B)
16	13, 14, 15	3bitr4i 268	. 2 ⊢ (y ∈ ran {⟨A, B⟩} ↔ y ∈ {B})
17	16	eqriv 2350	1 ⊢ ran {⟨A, B⟩} = {B}

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {csn 3737  ⟨cop 4561   class class class wbr 4639  ran crn 4773

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 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-br 4640  df-ima 4727  df-rn 4786

This theorem is referenced by:  op2nda  5076  fpr  5437  1cnc  6139