keridl - Mathbox for Jeff Madsen

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Theorem keridl 33831

Description: The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)

**Hypotheses**
Ref	Expression
keridl.1	⊢ 𝐺 = (1^st ‘𝑆)
keridl.2	⊢ 𝑍 = (GId‘𝐺)

**Assertion**
Ref	Expression
keridl	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (^◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅))

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2622	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3		keridl.1	. . . 4 ⊢ 𝐺 = (1^st ‘𝑆)
4		eqid 2622	. . . 4 ⊢ ran 𝐺 = ran 𝐺
5	1, 2, 3, 4	rngohomf 33765	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran (1^st ‘𝑅)⟶ran 𝐺)
6		cnvimass 5485	. . . 4 ⊢ (^◡𝐹 “ {𝑍}) ⊆ dom 𝐹
7		fdm 6051	. . . 4 ⊢ (𝐹:ran (1^st ‘𝑅)⟶ran 𝐺 → dom 𝐹 = ran (1^st ‘𝑅))
8	6, 7	syl5sseq 3653	. . 3 ⊢ (𝐹:ran (1^st ‘𝑅)⟶ran 𝐺 → (^◡𝐹 “ {𝑍}) ⊆ ran (1^st ‘𝑅))
9	5, 8	syl 17	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (^◡𝐹 “ {𝑍}) ⊆ ran (1^st ‘𝑅))
10		eqid 2622	. . . . 5 ⊢ (GId‘(1^st ‘𝑅)) = (GId‘(1^st ‘𝑅))
11	1, 2, 10	rngo0cl 33718	. . . 4 ⊢ (𝑅 ∈ RingOps → (GId‘(1^st ‘𝑅)) ∈ ran (1^st ‘𝑅))
12	11	3ad2ant1 1082	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (GId‘(1^st ‘𝑅)) ∈ ran (1^st ‘𝑅))
13		keridl.2	. . . . 5 ⊢ 𝑍 = (GId‘𝐺)
14	1, 10, 3, 13	rngohom0 33771	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(1^st ‘𝑅))) = 𝑍)
15		fvex 6201	. . . . 5 ⊢ (𝐹‘(GId‘(1^st ‘𝑅))) ∈ V
16	15	elsn 4192	. . . 4 ⊢ ((𝐹‘(GId‘(1^st ‘𝑅))) ∈ {𝑍} ↔ (𝐹‘(GId‘(1^st ‘𝑅))) = 𝑍)
17	14, 16	sylibr 224	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(1^st ‘𝑅))) ∈ {𝑍})
18		ffn 6045	. . . 4 ⊢ (𝐹:ran (1^st ‘𝑅)⟶ran 𝐺 → 𝐹 Fn ran (1^st ‘𝑅))
19		elpreima 6337	. . . 4 ⊢ (𝐹 Fn ran (1^st ‘𝑅) → ((GId‘(1^st ‘𝑅)) ∈ (^◡𝐹 “ {𝑍}) ↔ ((GId‘(1^st ‘𝑅)) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(GId‘(1^st ‘𝑅))) ∈ {𝑍})))
20	5, 18, 19	3syl 18	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((GId‘(1^st ‘𝑅)) ∈ (^◡𝐹 “ {𝑍}) ↔ ((GId‘(1^st ‘𝑅)) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(GId‘(1^st ‘𝑅))) ∈ {𝑍})))
21	12, 17, 20	mpbir2and 957	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (GId‘(1^st ‘𝑅)) ∈ (^◡𝐹 “ {𝑍}))
22		an4 865	. . . . . . . 8 ⊢ (((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑦) ∈ {𝑍})) ↔ ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) ∧ ((𝐹‘𝑥) ∈ {𝑍} ∧ (𝐹‘𝑦) ∈ {𝑍})))
23	1, 2, 3	rngohomadd 33768	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)𝐺(𝐹‘𝑦)))
24	23	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) ∧ ((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍)) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)𝐺(𝐹‘𝑦)))
25		oveq12 6659	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍) → ((𝐹‘𝑥)𝐺(𝐹‘𝑦)) = (𝑍𝐺𝑍))
26	25	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) ∧ ((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍)) → ((𝐹‘𝑥)𝐺(𝐹‘𝑦)) = (𝑍𝐺𝑍))
27	3	rngogrpo 33709	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ RingOps → 𝐺 ∈ GrpOp)
28	4, 13	grpoidcl 27368	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ ran 𝐺)
29	4, 13	grpolid 27370	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
30	28, 29	mpdan 702	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ GrpOp → (𝑍𝐺𝑍) = 𝑍)
31	27, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
32	31	3ad2ant2 1083	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝑍𝐺𝑍) = 𝑍)
33	32	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) ∧ ((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍)) → (𝑍𝐺𝑍) = 𝑍)
34	24, 26, 33	3eqtrd 2660	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) ∧ ((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍)) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = 𝑍)
35	34	ex 450	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = 𝑍))
36		fvex 6201	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑥) ∈ V
37	36	elsn 4192	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍)
38		fvex 6201	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑦) ∈ V
39	38	elsn 4192	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑦) ∈ {𝑍} ↔ (𝐹‘𝑦) = 𝑍)
40	37, 39	anbi12i 733	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) ∈ {𝑍} ∧ (𝐹‘𝑦) ∈ {𝑍}) ↔ ((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍))
41		fvex 6201	. . . . . . . . . . . 12 ⊢ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ V
42	41	elsn 4192	. . . . . . . . . . 11 ⊢ ((𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍} ↔ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = 𝑍)
43	35, 40, 42	3imtr4g 285	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (((𝐹‘𝑥) ∈ {𝑍} ∧ (𝐹‘𝑦) ∈ {𝑍}) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍}))
44	43	imdistanda 729	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) ∧ ((𝐹‘𝑥) ∈ {𝑍} ∧ (𝐹‘𝑦) ∈ {𝑍})) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍})))
45	1, 2	rngogcl 33711	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅))
46	45	3expib 1268	. . . . . . . . . . 11 ⊢ (𝑅 ∈ RingOps → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅)))
47	46	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅)))
48	47	anim1d 588	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍}) → ((𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍})))
49	44, 48	syld 47	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) ∧ ((𝐹‘𝑥) ∈ {𝑍} ∧ (𝐹‘𝑦) ∈ {𝑍})) → ((𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍})))
50	22, 49	syl5bi 232	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑦) ∈ {𝑍})) → ((𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍})))
51		elpreima 6337	. . . . . . . . 9 ⊢ (𝐹 Fn ran (1^st ‘𝑅) → (𝑥 ∈ (^◡𝐹 “ {𝑍}) ↔ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍})))
52	5, 18, 51	3syl 18	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝑥 ∈ (^◡𝐹 “ {𝑍}) ↔ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍})))
53		elpreima 6337	. . . . . . . . 9 ⊢ (𝐹 Fn ran (1^st ‘𝑅) → (𝑦 ∈ (^◡𝐹 “ {𝑍}) ↔ (𝑦 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑦) ∈ {𝑍})))
54	5, 18, 53	3syl 18	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝑦 ∈ (^◡𝐹 “ {𝑍}) ↔ (𝑦 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑦) ∈ {𝑍})))
55	52, 54	anbi12d 747	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ (^◡𝐹 “ {𝑍}) ∧ 𝑦 ∈ (^◡𝐹 “ {𝑍})) ↔ ((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑦) ∈ {𝑍}))))
56		elpreima 6337	. . . . . . . 8 ⊢ (𝐹 Fn ran (1^st ‘𝑅) → ((𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍})))
57	5, 18, 56	3syl 18	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍})))
58	50, 55, 57	3imtr4d 283	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ (^◡𝐹 “ {𝑍}) ∧ 𝑦 ∈ (^◡𝐹 “ {𝑍})) → (𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍})))
59	58	impl 650	. . . . 5 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ (^◡𝐹 “ {𝑍})) ∧ 𝑦 ∈ (^◡𝐹 “ {𝑍})) → (𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}))
60	59	ralrimiva 2966	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ (^◡𝐹 “ {𝑍})) → ∀𝑦 ∈ (^◡𝐹 “ {𝑍})(𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}))
61	37	anbi2i 730	. . . . . . 7 ⊢ ((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍}) ↔ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍))
62		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
63	1, 62, 2	rngocl 33700	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1^st ‘𝑅) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) → (𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅))
64	63	3expb 1266	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ RingOps ∧ (𝑧 ∈ ran (1^st ‘𝑅) ∧ 𝑥 ∈ ran (1^st ‘𝑅))) → (𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅))
65	64	3ad2antl1 1223	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑧 ∈ ran (1^st ‘𝑅) ∧ 𝑥 ∈ ran (1^st ‘𝑅))) → (𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅))
66	65	anass1rs 849	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅))
67	66	adantlrr 757	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅))
68		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (2^nd ‘𝑆) = (2^nd ‘𝑆)
69	1, 2, 62, 68	rngohommul 33769	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑧 ∈ ran (1^st ‘𝑅) ∧ 𝑥 ∈ ran (1^st ‘𝑅))) → (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) = ((𝐹‘𝑧)(2^nd ‘𝑆)(𝐹‘𝑥)))
70	69	anass1rs 849	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) = ((𝐹‘𝑧)(2^nd ‘𝑆)(𝐹‘𝑥)))
71	70	adantlrr 757	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) = ((𝐹‘𝑧)(2^nd ‘𝑆)(𝐹‘𝑥)))
72		oveq2 6658	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) = 𝑍 → ((𝐹‘𝑧)(2^nd ‘𝑆)(𝐹‘𝑥)) = ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍))
73	72	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍) → ((𝐹‘𝑧)(2^nd ‘𝑆)(𝐹‘𝑥)) = ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍))
74	73	ad2antlr 763	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝐹‘𝑧)(2^nd ‘𝑆)(𝐹‘𝑥)) = ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍))
75	1, 2, 3, 4	rngohomcl 33766	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘𝑧) ∈ ran 𝐺)
76	13, 4, 3, 68	rngorz 33722	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ RingOps ∧ (𝐹‘𝑧) ∈ ran 𝐺) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
77	76	3ad2antl2 1224	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹‘𝑧) ∈ ran 𝐺) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
78	75, 77	syldan 487	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
79	78	adantlr 751	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
80	71, 74, 79	3eqtrd 2660	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) = 𝑍)
81		fvex 6201	. . . . . . . . . . . . 13 ⊢ (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ V
82	81	elsn 4192	. . . . . . . . . . . 12 ⊢ ((𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ {𝑍} ↔ (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) = 𝑍)
83	80, 82	sylibr 224	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ {𝑍})
84		elpreima 6337	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn ran (1^st ‘𝑅) → ((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ {𝑍})))
85	5, 18, 84	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ {𝑍})))
86	85	ad2antrr 762	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ {𝑍})))
87	67, 83, 86	mpbir2and 957	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}))
88	1, 62, 2	rngocl 33700	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅))
89	88	3expb 1266	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑧 ∈ ran (1^st ‘𝑅))) → (𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅))
90	89	3ad2antl1 1223	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑧 ∈ ran (1^st ‘𝑅))) → (𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅))
91	90	anassrs 680	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅))
92	91	adantlrr 757	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅))
93	1, 2, 62, 68	rngohommul 33769	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑧 ∈ ran (1^st ‘𝑅))) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑧)))
94	93	anassrs 680	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑧)))
95	94	adantlrr 757	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑧)))
96		oveq1 6657	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) = 𝑍 → ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑧)) = (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)))
97	96	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍) → ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑧)) = (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)))
98	97	ad2antlr 763	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑧)) = (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)))
99	13, 4, 3, 68	rngolz 33721	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ RingOps ∧ (𝐹‘𝑧) ∈ ran 𝐺) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
100	99	3ad2antl2 1224	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹‘𝑧) ∈ ran 𝐺) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
101	75, 100	syldan 487	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
102	101	adantlr 751	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
103	95, 98, 102	3eqtrd 2660	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) = 𝑍)
104		fvex 6201	. . . . . . . . . . . . 13 ⊢ (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ V
105	104	elsn 4192	. . . . . . . . . . . 12 ⊢ ((𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ {𝑍} ↔ (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) = 𝑍)
106	103, 105	sylibr 224	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ {𝑍})
107		elpreima 6337	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn ran (1^st ‘𝑅) → ((𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ {𝑍})))
108	5, 18, 107	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ {𝑍})))
109	108	ad2antrr 762	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ {𝑍})))
110	92, 106, 109	mpbir2and 957	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))
111	87, 110	jca 554	. . . . . . . . 9 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍})))
112	111	ralrimiva 2966	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) → ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍})))
113	112	ex 450	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍) → ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))
114	61, 113	syl5bi 232	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍}) → ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))
115	52, 114	sylbid 230	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝑥 ∈ (^◡𝐹 “ {𝑍}) → ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))
116	115	imp 445	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ (^◡𝐹 “ {𝑍})) → ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍})))
117	60, 116	jca 554	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ (^◡𝐹 “ {𝑍})) → (∀𝑦 ∈ (^◡𝐹 “ {𝑍})(𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}) ∧ ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))
118	117	ralrimiva 2966	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ (^◡𝐹 “ {𝑍})(∀𝑦 ∈ (^◡𝐹 “ {𝑍})(𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}) ∧ ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))
119	1, 62, 2, 10	isidl 33813	. . 3 ⊢ (𝑅 ∈ RingOps → ((^◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅) ↔ ((^◡𝐹 “ {𝑍}) ⊆ ran (1^st ‘𝑅) ∧ (GId‘(1^st ‘𝑅)) ∈ (^◡𝐹 “ {𝑍}) ∧ ∀𝑥 ∈ (^◡𝐹 “ {𝑍})(∀𝑦 ∈ (^◡𝐹 “ {𝑍})(𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}) ∧ ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))))
120	119	3ad2ant1 1082	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((^◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅) ↔ ((^◡𝐹 “ {𝑍}) ⊆ ran (1^st ‘𝑅) ∧ (GId‘(1^st ‘𝑅)) ∈ (^◡𝐹 “ {𝑍}) ∧ ∀𝑥 ∈ (^◡𝐹 “ {𝑍})(∀𝑦 ∈ (^◡𝐹 “ {𝑍})(𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}) ∧ ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))))
121	9, 21, 118, 120	mpbir3and 1245	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (^◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 {csn 4177 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 “ cima 5117 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 GrpOpcgr 27343 GIdcgi 27344 RingOpscrngo 33693 RngHom crnghom 33759 Idlcidl 33806

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-grpo 27347 df-gid 27348 df-ginv 27349 df-ablo 27399 df-ghomOLD 33683 df-rngo 33694 df-rngohom 33762 df-idl 33809

This theorem is referenced by: (None)

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