Theorem brimage 5794

Description: Binary relationship over the image function. (Contributed by SF, 11-Feb-2015.)

Hypotheses

Ref	Expression
brimage.1	⊢ A ∈ V
brimage.2	⊢ B ∈ V

Assertion

Ref	Expression
brimage	⊢ (AImageRB ↔ B = (R “ A))

Proof of Theorem brimage

Dummy variables x y t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elima1c 4948	. . . 4 ⊢ (⟨A, B⟩ ∈ (( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)) “ 1c) ↔ ∃x⟨{x}, ⟨A, B⟩⟩ ∈ ( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)))
2		elsymdif 3224	. . . . . 6 ⊢ (⟨{x}, ⟨A, B⟩⟩ ∈ ( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)) ↔ ¬ (⟨{x}, ⟨A, B⟩⟩ ∈ Ins2 S ↔ ⟨{x}, ⟨A, B⟩⟩ ∈ Ins3 ( S ∘ ◡ SI R)))
3		brimage.1	. . . . . . . . 9 ⊢ A ∈ V
4	3	otelins2 5792	. . . . . . . 8 ⊢ (⟨{x}, ⟨A, B⟩⟩ ∈ Ins2 S ↔ ⟨{x}, B⟩ ∈ S )
5		vex 2863	. . . . . . . . 9 ⊢ x ∈ V
6		brimage.2	. . . . . . . . 9 ⊢ B ∈ V
7	5, 6	opelssetsn 4761	. . . . . . . 8 ⊢ (⟨{x}, B⟩ ∈ S ↔ x ∈ B)
8	4, 7	bitri 240	. . . . . . 7 ⊢ (⟨{x}, ⟨A, B⟩⟩ ∈ Ins2 S ↔ x ∈ B)
9	6	otelins3 5793	. . . . . . . 8 ⊢ (⟨{x}, ⟨A, B⟩⟩ ∈ Ins3 ( S ∘ ◡ SI R) ↔ ⟨{x}, A⟩ ∈ ( S ∘ ◡ SI R))
10		brcnv 4893	. . . . . . . . . . . . . 14 ⊢ ({x}◡ SI Rt ↔ t SI R{x})
11	5	brsnsi2 5777	. . . . . . . . . . . . . 14 ⊢ (t SI R{x} ↔ ∃y(t = {y} ∧ yRx))
12	10, 11	bitri 240	. . . . . . . . . . . . 13 ⊢ ({x}◡ SI Rt ↔ ∃y(t = {y} ∧ yRx))
13	12	anbi1i 676	. . . . . . . . . . . 12 ⊢ (({x}◡ SI Rt ∧ t S A) ↔ (∃y(t = {y} ∧ yRx) ∧ t S A))
14		19.41v 1901	. . . . . . . . . . . 12 ⊢ (∃y((t = {y} ∧ yRx) ∧ t S A) ↔ (∃y(t = {y} ∧ yRx) ∧ t S A))
15	13, 14	bitr4i 243	. . . . . . . . . . 11 ⊢ (({x}◡ SI Rt ∧ t S A) ↔ ∃y((t = {y} ∧ yRx) ∧ t S A))
16	15	exbii 1582	. . . . . . . . . 10 ⊢ (∃t({x}◡ SI Rt ∧ t S A) ↔ ∃t∃y((t = {y} ∧ yRx) ∧ t S A))
17		excom 1741	. . . . . . . . . 10 ⊢ (∃t∃y((t = {y} ∧ yRx) ∧ t S A) ↔ ∃y∃t((t = {y} ∧ yRx) ∧ t S A))
18		anass 630	. . . . . . . . . . . . 13 ⊢ (((t = {y} ∧ yRx) ∧ t S A) ↔ (t = {y} ∧ (yRx ∧ t S A)))
19	18	exbii 1582	. . . . . . . . . . . 12 ⊢ (∃t((t = {y} ∧ yRx) ∧ t S A) ↔ ∃t(t = {y} ∧ (yRx ∧ t S A)))
20		snex 4112	. . . . . . . . . . . . 13 ⊢ {y} ∈ V
21		breq1 4643	. . . . . . . . . . . . . . 15 ⊢ (t = {y} → (t S A ↔ {y} S A))
22	21	anbi2d 684	. . . . . . . . . . . . . 14 ⊢ (t = {y} → ((yRx ∧ t S A) ↔ (yRx ∧ {y} S A)))
23		ancom 437	. . . . . . . . . . . . . . 15 ⊢ ((yRx ∧ {y} S A) ↔ ({y} S A ∧ yRx))
24		vex 2863	. . . . . . . . . . . . . . . . 17 ⊢ y ∈ V
25	24, 3	brssetsn 4760	. . . . . . . . . . . . . . . 16 ⊢ ({y} S A ↔ y ∈ A)
26	25	anbi1i 676	. . . . . . . . . . . . . . 15 ⊢ (({y} S A ∧ yRx) ↔ (y ∈ A ∧ yRx))
27	23, 26	bitri 240	. . . . . . . . . . . . . 14 ⊢ ((yRx ∧ {y} S A) ↔ (y ∈ A ∧ yRx))
28	22, 27	syl6bb 252	. . . . . . . . . . . . 13 ⊢ (t = {y} → ((yRx ∧ t S A) ↔ (y ∈ A ∧ yRx)))
29	20, 28	ceqsexv 2895	. . . . . . . . . . . 12 ⊢ (∃t(t = {y} ∧ (yRx ∧ t S A)) ↔ (y ∈ A ∧ yRx))
30	19, 29	bitri 240	. . . . . . . . . . 11 ⊢ (∃t((t = {y} ∧ yRx) ∧ t S A) ↔ (y ∈ A ∧ yRx))
31	30	exbii 1582	. . . . . . . . . 10 ⊢ (∃y∃t((t = {y} ∧ yRx) ∧ t S A) ↔ ∃y(y ∈ A ∧ yRx))
32	16, 17, 31	3bitri 262	. . . . . . . . 9 ⊢ (∃t({x}◡ SI Rt ∧ t S A) ↔ ∃y(y ∈ A ∧ yRx))
33		opelco 4885	. . . . . . . . 9 ⊢ (⟨{x}, A⟩ ∈ ( S ∘ ◡ SI R) ↔ ∃t({x}◡ SI Rt ∧ t S A))
34		elima2 4756	. . . . . . . . 9 ⊢ (x ∈ (R “ A) ↔ ∃y(y ∈ A ∧ yRx))
35	32, 33, 34	3bitr4i 268	. . . . . . . 8 ⊢ (⟨{x}, A⟩ ∈ ( S ∘ ◡ SI R) ↔ x ∈ (R “ A))
36	9, 35	bitri 240	. . . . . . 7 ⊢ (⟨{x}, ⟨A, B⟩⟩ ∈ Ins3 ( S ∘ ◡ SI R) ↔ x ∈ (R “ A))
37	8, 36	bibi12i 306	. . . . . 6 ⊢ ((⟨{x}, ⟨A, B⟩⟩ ∈ Ins2 S ↔ ⟨{x}, ⟨A, B⟩⟩ ∈ Ins3 ( S ∘ ◡ SI R)) ↔ (x ∈ B ↔ x ∈ (R “ A)))
38	2, 37	xchbinx 301	. . . . 5 ⊢ (⟨{x}, ⟨A, B⟩⟩ ∈ ( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)) ↔ ¬ (x ∈ B ↔ x ∈ (R “ A)))
39	38	exbii 1582	. . . 4 ⊢ (∃x⟨{x}, ⟨A, B⟩⟩ ∈ ( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)) ↔ ∃x ¬ (x ∈ B ↔ x ∈ (R “ A)))
40		exnal 1574	. . . 4 ⊢ (∃x ¬ (x ∈ B ↔ x ∈ (R “ A)) ↔ ¬ ∀x(x ∈ B ↔ x ∈ (R “ A)))
41	1, 39, 40	3bitri 262	. . 3 ⊢ (⟨A, B⟩ ∈ (( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)) “ 1c) ↔ ¬ ∀x(x ∈ B ↔ x ∈ (R “ A)))
42	41	con2bii 322	. 2 ⊢ (∀x(x ∈ B ↔ x ∈ (R “ A)) ↔ ¬ ⟨A, B⟩ ∈ (( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)) “ 1c))
43		dfcleq 2347	. 2 ⊢ (B = (R “ A) ↔ ∀x(x ∈ B ↔ x ∈ (R “ A)))
44		df-image 5755	. . . 4 ⊢ ImageR = ∼ (( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)) “ 1c)
45	44	breqi 4646	. . 3 ⊢ (AImageRB ↔ A ∼ (( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)) “ 1c)B)
46		df-br 4641	. . 3 ⊢ (A ∼ (( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)) “ 1c)B ↔ ⟨A, B⟩ ∈ ∼ (( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)) “ 1c))
47	3, 6	opex 4589	. . . 4 ⊢ ⟨A, B⟩ ∈ V
48	47	elcompl 3226	. . 3 ⊢ (⟨A, B⟩ ∈ ∼ (( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)) “ 1c) ↔ ¬ ⟨A, B⟩ ∈ (( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)) “ 1c))
49	45, 46, 48	3bitri 262	. 2 ⊢ (AImageRB ↔ ¬ ⟨A, B⟩ ∈ (( Ins2 S ⊕ Ins3 ( S ∘ ◡ SI R)) “ 1c))
50	42, 43, 49	3bitr4ri 269	1 ⊢ (AImageRB ↔ B = (R “ A))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∼ ccompl 3206   ⊕ csymdif 3210  {csn 3738  1_cc1c 4135  ⟨cop 4562   class class class wbr 4640   S csset 4720   SI csi 4721   ∘ ccom 4722   “ cima 4723  ^◡ccnv 4772   Ins2 cins2 5750   Ins3 cins3 5752  Imagecimage 5754

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-cnv 4786  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755

This theorem is used by:  fnsex  5833  clos1ex  5877  mapexi  6004  enex  6032  ovcelem1  6172  ceex  6175  nchoicelem10  6299