dfima2 - New Foundations Explorer

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Theorem dfima2 4746

Description: Express the image functor in terms of the set construction functions. (Contributed by SF, 7-Jan-2015.)

**Assertion**
Ref	Expression
dfima2	⊢ (A “ B) = ((((V ×_k (V ×_k V)) ∩ ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c))) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁A) “_k B)

Proof of Theorem dfima2 Dummy variables x y z t w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ima 4728	. 2 ⊢ (A “ B) = {y ∣ ∃x ∈ B xAy}
2		vex 2863	. . . . 5 ⊢ y ∈ V
3	2	elimak 4260	. . . 4 ⊢ (y ∈ ((((V ×_k (V ×_k V)) ∩ ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c))) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁A) “_k B) ↔ ∃x ∈ B ⟪x, y⟫ ∈ (((V ×_k (V ×_k V)) ∩ ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c))) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁A))
4		setconslem6 4737	. . . . . . 7 ⊢ (((V ×_k (V ×_k V)) ∩ ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c))) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁A) = {z ∣ ∃w∃t(z = ⟪w, t⟫ ∧ ⟨w, t⟩ ∈ A)}
5		opeq1 4579	. . . . . . . 8 ⊢ (w = x → ⟨w, t⟩ = ⟨x, t⟩)
6	5	eleq1d 2419	. . . . . . 7 ⊢ (w = x → (⟨w, t⟩ ∈ A ↔ ⟨x, t⟩ ∈ A))
7		opeq2 4580	. . . . . . . 8 ⊢ (t = y → ⟨x, t⟩ = ⟨x, y⟩)
8	7	eleq1d 2419	. . . . . . 7 ⊢ (t = y → (⟨x, t⟩ ∈ A ↔ ⟨x, y⟩ ∈ A))
9		vex 2863	. . . . . . 7 ⊢ x ∈ V
10	4, 6, 8, 9, 2	opkelopkab 4247	. . . . . 6 ⊢ (⟪x, y⟫ ∈ (((V ×_k (V ×_k V)) ∩ ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c))) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁A) ↔ ⟨x, y⟩ ∈ A)
11		df-br 4641	. . . . . 6 ⊢ (xAy ↔ ⟨x, y⟩ ∈ A)
12	10, 11	bitr4i 243	. . . . 5 ⊢ (⟪x, y⟫ ∈ (((V ×_k (V ×_k V)) ∩ ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c))) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁A) ↔ xAy)
13	12	rexbii 2640	. . . 4 ⊢ (∃x ∈ B ⟪x, y⟫ ∈ (((V ×_k (V ×_k V)) ∩ ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c))) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁A) ↔ ∃x ∈ B xAy)
14	3, 13	bitri 240	. . 3 ⊢ (y ∈ ((((V ×_k (V ×_k V)) ∩ ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c))) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁A) “_k B) ↔ ∃x ∈ B xAy)
15	14	eqabi 2465	. 2 ⊢ ((((V ×_k (V ×_k V)) ∩ ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c))) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁A) “_k B) = {y ∣ ∃x ∈ B xAy}
16	1, 15	eqtr4i 2376	1 ⊢ (A “ B) = ((((V ×_k (V ×_k V)) ∩ ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k 𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c))) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁A) “_k B)

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1642 ∈ wcel 1710 {cab 2339 ∃wrex 2616 Vcvv 2860 ∼ ccompl 3206 ∖ cdif 3207 ∪ cun 3208 ∩ cin 3209 ⊕ csymdif 3210 {csn 3738 ⟪copk 4058 1_cc1c 4135 𝒫₁cpw1 4136 ×_k cxpk 4175 ^◡_kccnvk 4176 Ins2_k cins2k 4177 Ins3_k cins3k 4178 “_k cimak 4180 ∘_k ccomk 4181 SI_k csik 4182 Image_kcimagek 4183 S_k cssetk 4184 I_k cidk 4185 Nn cnnc 4374 0_cc0c 4375 ⟨cop 4562 class class class wbr 4640 “ cima 4723

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-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-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-rex 2621 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-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-addc 4379 df-nnc 4380 df-phi 4566 df-op 4567 df-br 4641 df-ima 4728

This theorem is used by: imaexg 4747

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