imakexg - New Foundations Explorer

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Theorem imakexg 4300

Description: The image of a set under a set is a set. (Contributed by SF, 14-Jan-2015.)

**Assertion**
Ref	Expression
imakexg	⊢ ((A ∈ V ∧ B ∈ W) → (A “_k B) ∈ V)

**Proof of Theorem imakexg**
Step	Hyp	Ref	Expression
1		dfimak2 4299	. 2 ⊢ (A “_k B) = ∼ P6 ( ∼ (1_c ×_k V) ∪ SI_k ∼ (A ∩ (B ×_k V)))
2		1cex 4143	. . . . . 6 ⊢ 1_c ∈ V
3		vvex 4110	. . . . . 6 ⊢ V ∈ V
4	2, 3	xpkex 4290	. . . . 5 ⊢ (1_c ×_k V) ∈ V
5	4	complex 4105	. . . 4 ⊢ ∼ (1_c ×_k V) ∈ V
6		xpkexg 4289	. . . . . . 7 ⊢ ((B ∈ W ∧ V ∈ V) → (B ×_k V) ∈ V)
7	3, 6	mpan2 652	. . . . . 6 ⊢ (B ∈ W → (B ×_k V) ∈ V)
8		inexg 4101	. . . . . 6 ⊢ ((A ∈ V ∧ (B ×_k V) ∈ V) → (A ∩ (B ×_k V)) ∈ V)
9	7, 8	sylan2 460	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ W) → (A ∩ (B ×_k V)) ∈ V)
10		complexg 4100	. . . . 5 ⊢ ((A ∩ (B ×_k V)) ∈ V → ∼ (A ∩ (B ×_k V)) ∈ V)
11		sikexg 4297	. . . . 5 ⊢ ( ∼ (A ∩ (B ×_k V)) ∈ V → SI_k ∼ (A ∩ (B ×_k V)) ∈ V)
12	9, 10, 11	3syl 18	. . . 4 ⊢ ((A ∈ V ∧ B ∈ W) → SI_k ∼ (A ∩ (B ×_k V)) ∈ V)
13		unexg 4102	. . . 4 ⊢ (( ∼ (1_c ×_k V) ∈ V ∧ SI_k ∼ (A ∩ (B ×_k V)) ∈ V) → ( ∼ (1_c ×_k V) ∪ SI_k ∼ (A ∩ (B ×_k V))) ∈ V)
14	5, 12, 13	sylancr 644	. . 3 ⊢ ((A ∈ V ∧ B ∈ W) → ( ∼ (1_c ×_k V) ∪ SI_k ∼ (A ∩ (B ×_k V))) ∈ V)
15		p6exg 4291	. . 3 ⊢ (( ∼ (1_c ×_k V) ∪ SI_k ∼ (A ∩ (B ×_k V))) ∈ V → P6 ( ∼ (1_c ×_k V) ∪ SI_k ∼ (A ∩ (B ×_k V))) ∈ V)
16		complexg 4100	. . 3 ⊢ ( P6 ( ∼ (1_c ×_k V) ∪ SI_k ∼ (A ∩ (B ×_k V))) ∈ V → ∼ P6 ( ∼ (1_c ×_k V) ∪ SI_k ∼ (A ∩ (B ×_k V))) ∈ V)
17	14, 15, 16	3syl 18	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → ∼ P6 ( ∼ (1_c ×_k V) ∪ SI_k ∼ (A ∩ (B ×_k V))) ∈ V)
18	1, 17	syl5eqel 2437	1 ⊢ ((A ∈ V ∧ B ∈ W) → (A “_k B) ∈ V)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 358 ∈ wcel 1710 Vcvv 2860 ∼ ccompl 3206 ∪ cun 3208 ∩ cin 3209 1_cc1c 4135 ×_k cxpk 4175 P6 cp6 4179 “_k cimak 4180 SI_k csik 4182

This theorem was proved from 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-si 4084 ax-typlower 4087 ax-sn 4088

This theorem 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-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-ss 3260 df-nul 3552 df-sn 3742 df-pr 3743 df-opk 4059 df-1c 4137 df-xpk 4186 df-cnvk 4187 df-imak 4190 df-p6 4192 df-sik 4193

This theorem is referenced by: imakex 4301 pw1exg 4303 cokexg 4310 imagekexg 4312 uniexg 4317 intexg 4320 pwexg 4329 addcexg 4394 phiexg 4572 opexg 4588 proj1exg 4592 proj2exg 4593 imaexg 4747 coexg 4750 siexg 4753

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