Theorem funcsetcestrclem1 16794

Description: Lemma 1 for funcsetcestrc 16804. (Contributed by AV, 27-Mar-2020.)

Hypotheses

Ref	Expression
funcsetcestrc.s	|- S = ( SetCat ` U )
funcsetcestrc.c	|- C = ( Base ` S )
funcsetcestrc.f	|- ( ph -> F = ( x e. C |-> { <. ( Base ` ndx ) , x >. } ) )

Assertion

Ref	Expression
funcsetcestrclem1	|- ( ( ph /\ X e. C ) -> ( F ` X ) = { <. ( Base ` ndx ) , X >. } )

Distinct variable groups:   x, C   x, X   ph, x

Allowed substitution hints:   S( x)   U( x)   F( x)

Proof of Theorem funcsetcestrclem1

Step	Hyp	Ref	Expression
1		funcsetcestrc.f	. . 3 |- ( ph -> F = ( x e. C |-> { <. ( Base ` ndx ) , x >. } ) )
2	1	adantr 481	. 2 |- ( ( ph /\ X e. C ) -> F = ( x e. C |-> { <. ( Base ` ndx ) , x >. } ) )
3		opeq2 4403	. . . 4 |- ( x = X -> <. ( Base ` ndx ) , x >. = <. ( Base ` ndx ) , X >. )
4	3	sneqd 4189	. . 3 |- ( x = X -> { <. ( Base ` ndx ) , x >. } = { <. ( Base ` ndx ) , X >. } )
5	4	adantl 482	. 2 |- ( ( ( ph /\ X e. C ) /\ x = X ) -> { <. ( Base ` ndx ) , x >. } = { <. ( Base ` ndx ) , X >. } )
6		simpr 477	. 2 |- ( ( ph /\ X e. C ) -> X e. C )
7		snex 4908	. . 3 |- { <. ( Base ` ndx ) , X >. } e. _V
8	7	a1i 11	. 2 |- ( ( ph /\ X e. C ) -> { <. ( Base ` ndx ) , X >. } e. _V )
9	2, 5, 6, 8	fvmptd 6288	1 |- ( ( ph /\ X e. C ) -> ( F ` X ) = { <. ( Base ` ndx ) , X >. } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   |-> cmpt 4729   ` cfv 5888   ndxcnx 15854   Basecbs 15857   SetCatcsetc 16725

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-csb 3534  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  funcsetcestrclem2  16795  embedsetcestrclem  16797  funcsetcestrclem7  16801  funcsetcestrclem8  16802  funcsetcestrclem9  16803  fullsetcestrc  16806