Theorem opabiotadm 6260

Description: Define a function whose value is "the unique y such that ph ( x , y )". (Contributed by NM, 16-Nov-2013.)

Hypothesis

Ref	Expression
opabiota.1	|- F = { <. x , y >. | { y | ph } = { y } }

Assertion

Ref	Expression
opabiotadm	|- dom F = { x | E! y ph }

Distinct variable group:   x, y, F

Allowed substitution hints:   ph( x, y)

Proof of Theorem opabiotadm

Step	Hyp	Ref	Expression
1		dmopab 5335	. 2 |- dom { <. x , y >. | { y | ph } = { y } } = { x | E. y { y | ph } = { y } }
2		opabiota.1	. . 3 |- F = { <. x , y >. | { y | ph } = { y } }
3	2	dmeqi 5325	. 2 |- dom F = dom { <. x , y >. | { y | ph } = { y } }
4		euabsn 4261	. . 3 |- ( E! y ph <-> E. y { y | ph } = { y } )
5	4	abbii 2739	. 2 |- { x | E! y ph } = { x | E. y { y | ph } = { y } }
6	1, 3, 5	3eqtr4i 2654	1 |- dom F = { x | E! y ph }

Syntax hints:   = wceq 1483   E.wex 1704   E!weu 2470   {cab 2608   {csn 4177   {copab 4712   dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-dm 5124

This theorem is referenced by:  opabiota  6261