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Theorem iunfopab 5204
 Description: Two ways to express a function as a class of ordered pairs. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Sep-2011.) (Contributed by set.mm contributors, 19-Dec-2008.)
Hypothesis
Ref Expression
iunfopab.1
Assertion
Ref Expression
iunfopab
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem iunfopab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-rex 2620 . . . 4
2 vex 2862 . . . . . . . 8
32elsnc 3756 . . . . . . 7
43anbi2i 675 . . . . . 6
5 iunfopab.1 . . . . . . 7
6 opeq2 4579 . . . . . . . . 9
76eqeq2d 2364 . . . . . . . 8
87anbi2d 684 . . . . . . 7
95, 8ceqsexv 2894 . . . . . 6
10 an13 774 . . . . . . 7
1110exbii 1582 . . . . . 6
124, 9, 113bitr2i 264 . . . . 5
1312exbii 1582 . . . 4
141, 13bitri 240 . . 3
1514abbii 2465 . 2
16 df-iun 3971 . 2
17 df-opab 4623 . 2
1815, 16, 173eqtr4i 2383 1
 Colors of variables: wff setvar class Syntax hints:   wa 358  wex 1541   wceq 1642   wcel 1710  cab 2339  wrex 2615  cvv 2859  csn 3737  ciun 3969  cop 4561  copab 4622 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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  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-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623 This theorem is referenced by: (None)
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