Theorem bnj1383 30902

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1383.1	|- ( ph <-> A. f e. A Fun f )
bnj1383.2	|- D = ( dom f i^i dom g )
bnj1383.3	|- ( ps <-> ( ph /\ A. f e. A A. g e. A ( f |` D ) = ( g |` D ) ) )

Assertion

Ref	Expression
bnj1383	|- ( ps -> Fun U. A )

Distinct variable groups:   A, f, g   ph, g

Allowed substitution hints:   ph( f)   ps( f, g)   D( f, g)

Proof of Theorem bnj1383

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1383.1	. 2 |- ( ph <-> A. f e. A Fun f )
2		bnj1383.2	. 2 |- D = ( dom f i^i dom g )
3		bnj1383.3	. 2 |- ( ps <-> ( ph /\ A. f e. A A. g e. A ( f |` D ) = ( g |` D ) ) )
4		biid 251	. 2 |- ( ( ps /\ <. x , y >. e. U. A /\ <. x , z >. e. U. A ) <-> ( ps /\ <. x , y >. e. U. A /\ <. x , z >. e. U. A ) )
5		biid 251	. 2 |- ( ( ( ps /\ <. x , y >. e. U. A /\ <. x , z >. e. U. A ) /\ f e. A /\ <. x , y >. e. f ) <-> ( ( ps /\ <. x , y >. e. U. A /\ <. x , z >. e. U. A ) /\ f e. A /\ <. x , y >. e. f ) )
6		biid 251	. 2 |- ( ( ( ( ps /\ <. x , y >. e. U. A /\ <. x , z >. e. U. A ) /\ f e. A /\ <. x , y >. e. f ) /\ g e. A /\ <. x , z >. e. g ) <-> ( ( ( ps /\ <. x , y >. e. U. A /\ <. x , z >. e. U. A ) /\ f e. A /\ <. x , y >. e. f ) /\ g e. A /\ <. x , z >. e. g ) )
7	1, 2, 3, 4, 5, 6	bnj1379 30901	1 |- ( ps -> Fun U. A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   <.cop 4183   U.cuni 4436   dom cdm 5114   |` cres 5116   Fun wfun 5882

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  bnj1385  30903  bnj60  31130