Theorem subvalt 5369

Description: Value of subtraction, which is the (unique) element x such that B + x = A. The notation U.{x e. CC | (B + x) = A} may at first seem cryptic but is actually a way of saying "the element x such that B + x = A" (see Theorem 8.17 of [Quine] p. 56); this works because there is only one such x as shown by negeu 5367, allowing us to exploit eusn 2450 and unisn 2521 (which you will find if you trace back the proof of subcl 5378).

Assertion

Ref	Expression
subvalt	|- ((A e. CC /\ B e. CC) -> (A - B) = U.{x e. CC | (B + x) = A})

Distinct variable groups:   x,A   x,B

Proof of Theorem subvalt

Step	Hyp	Ref	Expression
1		axcnex 5279	. . . 4 |- CC e. V
2	1	rabex 2730	. . 3 |- {x e. CC | (B + x) = A} e. V
3	2	uniex 2876	. 2 |- U.{x e. CC | (B + x) = A} e. V
4		eqeq2 1487	. . . 4 |- (y = A -> ((z + x) = y <-> (z + x) = A))
5	4	rabbisdv 1810	. . 3 |- (y = A -> {x e. CC | (z + x) = y} = {x e. CC | (z + x) = A})
6	5	unieqd 2516	. 2 |- (y = A -> U.{x e. CC | (z + x) = y} = U.{x e. CC | (z + x) = A})
7		opreq1 3974	. . . . 5 |- (z = B -> (z + x) = (B + x))
8	7	eqeq1d 1486	. . . 4 |- (z = B -> ((z + x) = A <-> (B + x) = A))
9	8	rabbisdv 1810	. . 3 |- (z = B -> {x e. CC | (z + x) = A} = {x e. CC | (B + x) = A})
10	9	unieqd 2516	. 2 |- (z = B -> U.{x e. CC | (z + x) = A} = U.{x e. CC | (B + x) = A})
11		df-sub 5368	. 2 |- - = {<.<.y, z>., w>. | ((y e. CC /\ z e. CC) /\ w = U.{x e. CC | (z + x) = y})}
12	3, 6, 10, 11	oprabval2 4034	1 |- ((A e. CC /\ B e. CC) -> (A - B) = U.{x e. CC | (B + x) = A})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  {crab 1651  U.cuni 2507  (class class class)co 3969  CCcc 5244   + caddc 5249   - cmin 5304

This theorem is referenced by:  subcl 5378  subopr 5382  subadd 5383  addinv 8124

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-opr 3971  df-oprab 3972  df-qs 4272  df-ni 5012  df-nq 5050  df-np 5098  df-nr 5179  df-c 5252  df-sub 5368

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