Theorem bnj142OLD 30794

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.) Obsolete as of 29-Dec-2018. This is now incorporated into the proof of fnsnb 6432.

Assertion

Ref	Expression
bnj142OLD	|- ( F Fn { A } -> ( u e. F -> u = <. A , ( F ` A ) >. ) )

Proof of Theorem bnj142OLD

Step	Hyp	Ref	Expression
1		fnresdm 6000	. . . 4 |- ( F Fn { A } -> ( F |` { A } ) = F )
2		fnfun 5988	. . . . 5 |- ( F Fn { A } -> Fun F )
3		funressn 6426	. . . . 5 |- ( Fun F -> ( F |` { A } ) C_ { <. A , ( F ` A ) >. } )
4	2, 3	syl 17	. . . 4 |- ( F Fn { A } -> ( F |` { A } ) C_ { <. A , ( F ` A ) >. } )
5	1, 4	eqsstr3d 3640	. . 3 |- ( F Fn { A } -> F C_ { <. A , ( F ` A ) >. } )
6	5	sseld 3602	. 2 |- ( F Fn { A } -> ( u e. F -> u e. { <. A , ( F ` A ) >. } ) )
7		elsni 4194	. 2 |- ( u e. { <. A , ( F ` A ) >. } -> u = <. A , ( F ` A ) >. )
8	6, 7	syl6 35	1 |- ( F Fn { A } -> ( u e. F -> u = <. A , ( F ` A ) >. ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   <.cop 4183   |` cres 5116   Fun wfun 5882   Fn wfn 5883   ` cfv 5888

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  bnj145OLD  30795