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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 B V
Assertion
Ref Expression
iunfopab x A {x, B} = {x, y (x A y = B)}
Distinct variable groups:   y,A   y,B   x,y
Allowed substitution hints:   A(x)   B(x)

Proof of Theorem iunfopab
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-rex 2620 . . . 4 (x A z {x, B} ↔ x(x A z {x, B}))
2 vex 2862 . . . . . . . 8 z V
32elsnc 3756 . . . . . . 7 (z {x, B} ↔ z = x, B)
43anbi2i 675 . . . . . 6 ((x A z {x, B}) ↔ (x A z = x, B))
5 iunfopab.1 . . . . . . 7 B V
6 opeq2 4579 . . . . . . . . 9 (y = Bx, y = x, B)
76eqeq2d 2364 . . . . . . . 8 (y = B → (z = x, yz = x, B))
87anbi2d 684 . . . . . . 7 (y = B → ((x A z = x, y) ↔ (x A z = x, B)))
95, 8ceqsexv 2894 . . . . . 6 (y(y = B (x A z = x, y)) ↔ (x A z = x, B))
10 an13 774 . . . . . . 7 ((y = B (x A z = x, y)) ↔ (z = x, y (x A y = B)))
1110exbii 1582 . . . . . 6 (y(y = B (x A z = x, y)) ↔ y(z = x, y (x A y = B)))
124, 9, 113bitr2i 264 . . . . 5 ((x A z {x, B}) ↔ y(z = x, y (x A y = B)))
1312exbii 1582 . . . 4 (x(x A z {x, B}) ↔ xy(z = x, y (x A y = B)))
141, 13bitri 240 . . 3 (x A z {x, B} ↔ xy(z = x, y (x A y = B)))
1514abbii 2465 . 2 {z x A z {x, B}} = {z xy(z = x, y (x A y = B))}
16 df-iun 3971 . 2 x A {x, B} = {z x A z {x, B}}
17 df-opab 4623 . 2 {x, y (x A y = B)} = {z xy(z = x, y (x A y = B))}
1815, 16, 173eqtr4i 2383 1 x A {x, B} = {x, y (x A y = B)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 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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