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Theorem rnoprab 5576
 Description: The range of an operation class abstraction. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Apr-2013.) (Contributed by set.mm contributors, 30-Aug-2004.) (Revised by set.mm contributors, 19-Apr-2013.)
Assertion
Ref Expression
rnoprab ran {x, y, z φ} = {z xyφ}
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem rnoprab
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 dfoprab2 5558 . . 3 {x, y, z φ} = {w, z xy(w = x, y φ)}
21rneqi 4957 . 2 ran {x, y, z φ} = ran {w, z xy(w = x, y φ)}
3 rnopab 4967 . 2 ran {w, z xy(w = x, y φ)} = {z wxy(w = x, y φ)}
4 exrot3 1744 . . . 4 (wxy(w = x, y φ) ↔ xyw(w = x, y φ))
5 19.41v 1901 . . . . . 6 (w(w = x, y φ) ↔ (w w = x, y φ))
6 vex 2862 . . . . . . . . . 10 x V
7 vex 2862 . . . . . . . . . 10 y V
86, 7opex 4588 . . . . . . . . 9 x, y V
98isseti 2865 . . . . . . . 8 w w = x, y
109biantrur 492 . . . . . . 7 (φ ↔ (w w = x, y φ))
1110bicomi 193 . . . . . 6 ((w w = x, y φ) ↔ φ)
125, 11bitri 240 . . . . 5 (w(w = x, y φ) ↔ φ)
13122exbii 1583 . . . 4 (xyw(w = x, y φ) ↔ xyφ)
144, 13bitri 240 . . 3 (wxy(w = x, y φ) ↔ xyφ)
1514abbii 2465 . 2 {z wxy(w = x, y φ)} = {z xyφ}
162, 3, 153eqtri 2377 1 ran {x, y, z φ} = {z xyφ}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642  {cab 2339  ⟨cop 4561  {copab 4622  ran crn 4773  {coprab 5527 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-ima 4727  df-rn 4786  df-oprab 5528 This theorem is referenced by:  rnoprab2  5577  rnmpt2  5717
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