Theorem funcringcsetcALTV2lem1 42036

Description: Lemma 1 for funcringcsetcALTV2 42045. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
funcringcsetcALTV2.r	|- R = (RingCat ` U )
funcringcsetcALTV2.s	|- S = ( SetCat ` U )
funcringcsetcALTV2.b	|- B = ( Base ` R )
funcringcsetcALTV2.c	|- C = ( Base ` S )
funcringcsetcALTV2.u	|- ( ph -> U e. WUni )
funcringcsetcALTV2.f	|- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )

Assertion

Ref	Expression
funcringcsetcALTV2lem1	|- ( ( ph /\ X e. B ) -> ( F ` X ) = ( Base ` X ) )

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

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

Proof of Theorem funcringcsetcALTV2lem1

Step	Hyp	Ref	Expression
1		funcringcsetcALTV2.f	. . 3 |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
2	1	adantr 481	. 2 |- ( ( ph /\ X e. B ) -> F = ( x e. B |-> ( Base ` x ) ) )
3		fveq2 6191	. . 3 |- ( x = X -> ( Base ` x ) = ( Base ` X ) )
4	3	adantl 482	. 2 |- ( ( ( ph /\ X e. B ) /\ x = X ) -> ( Base ` x ) = ( Base ` X ) )
5		simpr 477	. 2 |- ( ( ph /\ X e. B ) -> X e. B )
6		fvexd 6203	. 2 |- ( ( ph /\ X e. B ) -> ( Base ` X ) e. _V )
7	2, 4, 5, 6	fvmptd 6288	1 |- ( ( ph /\ X e. B ) -> ( F ` X ) = ( Base ` X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  WUnicwun 9522   Basecbs 15857   SetCatcsetc 16725  RingCatcringc 42003

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:  funcringcsetcALTV2lem2  42037  funcringcsetcALTV2lem7  42042  funcringcsetcALTV2lem8  42043  funcringcsetcALTV2lem9  42044