Theorem 1strwunbndx 15981

Description: A constructed one-slot structure in a weak universe containing the index of the base set extractor. (Contributed by AV, 27-Mar-2020.)

Hypotheses

Ref	Expression
1str.g	|- G = { <. ( Base ` ndx ) , B >. }
1strwun.u	|- ( ph -> U e. WUni )
1strwunbndx.b	|- ( ph -> ( Base ` ndx ) e. U )

Assertion

Ref	Expression
1strwunbndx	|- ( ( ph /\ B e. U ) -> G e. U )

Proof of Theorem 1strwunbndx

Step	Hyp	Ref	Expression
1		1str.g	. 2 |- G = { <. ( Base ` ndx ) , B >. }
2		1strwun.u	. . . 4 |- ( ph -> U e. WUni )
3	2	adantr 481	. . 3 |- ( ( ph /\ B e. U ) -> U e. WUni )
4		1strwunbndx.b	. . . . 5 |- ( ph -> ( Base ` ndx ) e. U )
5	4	adantr 481	. . . 4 |- ( ( ph /\ B e. U ) -> ( Base ` ndx ) e. U )
6		simpr 477	. . . 4 |- ( ( ph /\ B e. U ) -> B e. U )
7	3, 5, 6	wunop 9544	. . 3 |- ( ( ph /\ B e. U ) -> <. ( Base ` ndx ) , B >. e. U )
8	3, 7	wunsn 9538	. 2 |- ( ( ph /\ B e. U ) -> { <. ( Base ` ndx ) , B >. } e. U )
9	1, 8	syl5eqel 2705	1 |- ( ( ph /\ B e. U ) -> G e. U )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   ` cfv 5888  WUnicwun 9522   ndxcnx 15854   Basecbs 15857

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  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-uni 4437  df-tr 4753  df-wun 9524

This theorem is referenced by:  1strwun  15982  equivestrcsetc  16792