Theorem djulclr 7172

Description: Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)

Assertion

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

Proof of Theorem djulclr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvres 5618	. 2 |- ( C e. A -> ( (inl |` A ) ` C ) = (inl ` C ) )
2		elex 2785	. . . 4 |- ( C e. A -> C e. _V )
3		0ex 4182	. . . . . 6 |- (/) e. _V
4	3	snid 3669	. . . . 5 |- (/) e. { (/) }
5		opelxpi 4720	. . . . 5 |- ( ( (/) e. { (/) } /\ C e. A ) -> <. (/) , C >. e. ( { (/) } X. A ) )
6	4, 5	mpan 424	. . . 4 |- ( C e. A -> <. (/) , C >. e. ( { (/) } X. A ) )
7		opeq2 3829	. . . . 5 |- ( x = C -> <. (/) , x >. = <. (/) , C >. )
8		df-inl 7170	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
9	7, 8	fvmptg 5673	. . . 4 |- ( ( C e. _V /\ <. (/) , C >. e. ( { (/) } X. A ) ) -> (inl ` C ) = <. (/) , C >. )
10	2, 6, 9	syl2anc 411	. . 3 |- ( C e. A -> (inl ` C ) = <. (/) , C >. )
11		elun1 3344	. . . . 5 |- ( <. (/) , C >. e. ( { (/) } X. A ) -> <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
12	6, 11	syl 14	. . . 4 |- ( C e. A -> <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
13		df-dju 7161	. . . 4 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
14	12, 13	eleqtrrdi 2300	. . 3 |- ( C e. A -> <. (/) , C >. e. ( A ⊔ B ) )
15	10, 14	eqeltrd 2283	. 2 |- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
16	1, 15	eqeltrd 2283	1 |- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   _Vcvv 2773   u. cun 3168   (/)c0 3464   {csn 3638   <.cop 3641   X. cxp 4686   |` cres 4690   ` cfv 5285   1oc1o 6513   ⊔ cdju 7160  inlcinl 7168

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-res 4700  df-iota 5246  df-fun 5287  df-fv 5293  df-dju 7161  df-inl 7170

This theorem is referenced by:  inlresf1  7184