Theorem djulcl 7381

Description: Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)

Assertion

Ref	Expression
djulcl	|- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )

Proof of Theorem djulcl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2833	. . 3 |- ( C e. A -> C e. _V )
2		0ex 4255	. . . . 5 |- (/) e. _V
3	2	snid 3736	. . . 4 |- (/) e. { (/) }
4		opelxpi 4801	. . . 4 |- ( ( (/) e. { (/) } /\ C e. A ) -> <. (/) , C >. e. ( { (/) } X. A ) )
5	3, 4	mpan 428	. . 3 |- ( C e. A -> <. (/) , C >. e. ( { (/) } X. A ) )
6		opeq2 3900	. . . 4 |- ( x = C -> <. (/) , x >. = <. (/) , C >. )
7		df-inl 7377	. . . 4 |- inl = ( x e. _V |-> <. (/) , x >. )
8	6, 7	fvmptg 5775	. . 3 |- ( ( C e. _V /\ <. (/) , C >. e. ( { (/) } X. A ) ) -> (inl ` C ) = <. (/) , C >. )
9	1, 5, 8	syl2anc 415	. 2 |- ( C e. A -> (inl ` C ) = <. (/) , C >. )
10		elun1 3396	. . . 4 |- ( <. (/) , C >. e. ( { (/) } X. A ) -> <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
11	5, 10	syl 14	. . 3 |- ( C e. A -> <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
12		df-dju 7368	. . 3 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
13	11, 12	eleqtrrdi 2332	. 2 |- ( C e. A -> <. (/) , C >. e. ( A ⊔ B ) )
14	9, 13	eqeltrd 2315	1 |- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   (/)c0 3520   {csn 3705   <.cop 3708   X. cxp 4767   ` cfv 5372   1oc1o 6670   ⊔ cdju 7367  inlcinl 7375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-dju 7368  df-inl 7377

This theorem is referenced by:  djulclb  7385  updjudhcoinlf  7410  omp1eomlem  7424  difinfsnlem  7429  difinfsn  7430  ctmlemr  7438  ctm  7439  ctssdclemn0  7440  ctssdccl  7441  fodju0  7477  exmidfodomrlemr  7544  exmidfodomrlemrALT  7545  subctctexmid  16944