Theorem subvalt 5337

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 5335, allowing us to exploit eusn 2442 and unisn 2512 (which you will find if you trace back the proof of subcl 5346).

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 5247	. . . 4 |- CC e. V
2	1	rabex 2720	. . 3 |- {x e. CC | (B + x) = A} e. V
3	2	uniex 2865	. 2 |- U.{x e. CC | (B + x) = A} e. V
4		eqeq2 1481	. . . 4 |- (y = A -> ((z + x) = y <-> (z + x) = A))
5	4	rabbisdv 1803	. . 3 |- (y = A -> {x e. CC | (z + x) = y} = {x e. CC | (z + x) = A})
6	5	unieqd 2507	. 2 |- (y = A -> U.{x e. CC | (z + x) = y} = U.{x e. CC | (z + x) = A})
7		opreq1 3959	. . . . 5 |- (z = B -> (z + x) = (B + x))
8	7	eqeq1d 1480	. . . 4 |- (z = B -> ((z + x) = A <-> (B + x) = A))
9	8	rabbisdv 1803	. . 3 |- (z = B -> {x e. CC | (z + x) = A} = {x e. CC | (B + x) = A})
10	9	unieqd 2507	. 2 |- (z = B -> U.{x e. CC | (z + x) = A} = U.{x e. CC | (B + x) = A})
11		df-sub 5336	. 2 |- - = {<.<.y, z>., w>. | ((y e. CC /\ z e. CC) /\ w = U.{x e. CC | (z + x) = y})}
12	3, 6, 10, 11	oprabval2 4019	1 |- ((A e. CC /\ B e. CC) -> (A - B) = U.{x e. CC | (B + x) = A})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  {crab 1645  U.cuni 2498  (class class class)co 3954  CCcc 5212   + caddc 5217   - cmin 5272

This theorem is referenced by:  subcl 5346  subopr 5350  subadd 5351  addinv 8080

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-opr 3956  df-oprab 3957  df-qs 4256  df-ni 4980  df-nq 5018  df-np 5066  df-nr 5147  df-c 5220  df-sub 5336

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