Theorem rnopab 4967

Description: The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)

Assertion

Ref	Expression
rnopab	⊢ ran {⟨x, y⟩ ∣ φ} = {y ∣ ∃xφ}

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem rnopab

Step	Hyp	Ref	Expression
1		nfopab1 4628	. . 3 ⊢ Ⅎx{⟨x, y⟩ ∣ φ}
2		nfopab2 4629	. . 3 ⊢ Ⅎy{⟨x, y⟩ ∣ φ}
3	1, 2	dfrnf 4962	. 2 ⊢ ran {⟨x, y⟩ ∣ φ} = {y ∣ ∃x x{⟨x, y⟩ ∣ φ}y}
4		df-br 4640	. . . . 5 ⊢ (x{⟨x, y⟩ ∣ φ}y ↔ ⟨x, y⟩ ∈ {⟨x, y⟩ ∣ φ})
5		opabid 4695	. . . . 5 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ φ)
6	4, 5	bitri 240	. . . 4 ⊢ (x{⟨x, y⟩ ∣ φ}y ↔ φ)
7	6	exbii 1582	. . 3 ⊢ (∃x x{⟨x, y⟩ ∣ φ}y ↔ ∃xφ)
8	7	abbii 2465	. 2 ⊢ {y ∣ ∃x x{⟨x, y⟩ ∣ φ}y} = {y ∣ ∃xφ}
9	3, 8	eqtri 2373	1 ⊢ ran {⟨x, y⟩ ∣ φ} = {y ∣ ∃xφ}

Syntax hints:  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ⟨cop 4561  {copab 4622   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-opab 4623  df-br 4640  df-ima 4727  df-rn 4786

This theorem is referenced by:  rnopab2  4968  rnoprab  5576  mptpreima  5682  spacssnc  6284