Theorem wdom2d 8485

Description: Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 4771). (Contributed by Stefan O'Rear, 13-Feb-2015.)

Hypotheses

Ref	Expression
wdom2d.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
wdom2d.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
wdom2d.o	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)

Assertion

Ref	Expression
wdom2d	⊢ (𝜑 → 𝐴 ≼* 𝐵)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑋   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)   𝑋(𝑦)

Proof of Theorem wdom2d

Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wdom2d.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2		rabexg 4812	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ∈ V)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ∈ V)
4		wdom2d.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		xpexg 6960	. . . . 5 ⊢ (({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ∈ V ∧ 𝐴 ∈ 𝑉) → ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} × 𝐴) ∈ V)
6	3, 4, 5	syl2anc 693	. . . 4 ⊢ (𝜑 → ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} × 𝐴) ∈ V)
7		csbeq1 3536	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ⦋𝑧 / 𝑦⦌𝑋 = ⦋𝑤 / 𝑦⦌𝑋)
8	7	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴 ↔ ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴))
9	8	elrab 3363	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↔ (𝑤 ∈ 𝐵 ∧ ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴))
10	9	simprbi 480	. . . . . . 7 ⊢ (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} → ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴)
11	10	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}) → ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴)
12		eqid 2622	. . . . . 6 ⊢ (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋) = (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)
13	11, 12	fmptd 6385	. . . . 5 ⊢ (𝜑 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋):{𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}⟶𝐴)
14		fssxp 6060	. . . . 5 ⊢ ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋):{𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}⟶𝐴 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋) ⊆ ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} × 𝐴))
15	13, 14	syl 17	. . . 4 ⊢ (𝜑 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋) ⊆ ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} × 𝐴))
16	6, 15	ssexd 4805	. . 3 ⊢ (𝜑 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋) ∈ V)
17		wdom2d.o	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)
18		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
19	18	biimpcd 239	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑋 → 𝑋 ∈ 𝐴))
20	19	ancrd 577	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑋 → (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)))
21	20	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝑋 → (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)))
22	21	reximdv 3016	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝑥 = 𝑋 → ∃𝑦 ∈ 𝐵 (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)))
23	17, 22	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋))
24		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑣(𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)
25		nfcsb1v 3549	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝑋
26	25	nfel1 2779	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴
27	25	nfeq2 2780	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 = ⦋𝑣 / 𝑦⦌𝑋
28	26, 27	nfan 1828	. . . . . . . 8 ⊢ Ⅎ𝑦(⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋)
29		csbeq1a 3542	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → 𝑋 = ⦋𝑣 / 𝑦⦌𝑋)
30	29	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑋 ∈ 𝐴 ↔ ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴))
31	29	eqeq2d 2632	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑥 = 𝑋 ↔ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
32	30, 31	anbi12d 747	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) ↔ (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋)))
33	24, 28, 32	cbvrex 3168	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) ↔ ∃𝑣 ∈ 𝐵 (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
34	23, 33	sylib 208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑣 ∈ 𝐵 (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
35		csbeq1 3536	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑣 → ⦋𝑧 / 𝑦⦌𝑋 = ⦋𝑣 / 𝑦⦌𝑋)
36	35	eleq1d 2686	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑣 → (⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴 ↔ ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴))
37	36	elrab 3363	. . . . . . . . . . 11 ⊢ (𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↔ (𝑣 ∈ 𝐵 ∧ ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴))
38	37	simprbi 480	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} → ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴)
39		csbeq1 3536	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → ⦋𝑤 / 𝑦⦌𝑋 = ⦋𝑣 / 𝑦⦌𝑋)
40	39, 12	fvmptg 6280	. . . . . . . . . 10 ⊢ ((𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ∧ ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴) → ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)‘𝑣) = ⦋𝑣 / 𝑦⦌𝑋)
41	38, 40	mpdan 702	. . . . . . . . 9 ⊢ (𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} → ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)‘𝑣) = ⦋𝑣 / 𝑦⦌𝑋)
42	41	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} → (𝑥 = ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)‘𝑣) ↔ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
43	42	rexbiia 3040	. . . . . . 7 ⊢ (∃𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}𝑥 = ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)‘𝑣) ↔ ∃𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}𝑥 = ⦋𝑣 / 𝑦⦌𝑋)
44	36	rexrab 3370	. . . . . . 7 ⊢ (∃𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}𝑥 = ⦋𝑣 / 𝑦⦌𝑋 ↔ ∃𝑣 ∈ 𝐵 (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
45	43, 44	bitri 264	. . . . . 6 ⊢ (∃𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}𝑥 = ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)‘𝑣) ↔ ∃𝑣 ∈ 𝐵 (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
46	34, 45	sylibr 224	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}𝑥 = ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)‘𝑣))
47	46	ralrimiva 2966	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}𝑥 = ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)‘𝑣))
48		dffo3 6374	. . . 4 ⊢ ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋):{𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}–onto→𝐴 ↔ ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋):{𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}𝑥 = ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)‘𝑣)))
49	13, 47, 48	sylanbrc 698	. . 3 ⊢ (𝜑 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋):{𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}–onto→𝐴)
50		fowdom 8476	. . 3 ⊢ (((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋) ∈ V ∧ (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋):{𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}–onto→𝐴) → 𝐴 ≼* {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴})
51	16, 49, 50	syl2anc 693	. 2 ⊢ (𝜑 → 𝐴 ≼* {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴})
52		ssrab2 3687	. . . 4 ⊢ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ⊆ 𝐵
53		ssdomg 8001	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ⊆ 𝐵 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼ 𝐵))
54	52, 53	mpi 20	. . 3 ⊢ (𝐵 ∈ 𝑊 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼ 𝐵)
55		domwdom 8479	. . 3 ⊢ ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼ 𝐵 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼* 𝐵)
56	1, 54, 55	3syl 18	. 2 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼* 𝐵)
57		wdomtr 8480	. 2 ⊢ ((𝐴 ≼* {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ∧ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼* 𝐵) → 𝐴 ≼* 𝐵)
58	51, 56, 57	syl2anc 693	1 ⊢ (𝜑 → 𝐴 ≼* 𝐵)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200  ⦋csb 3533   ⊆ wss 3574   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888   ≼ cdom 7953   ≼^* cwdom 8462

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-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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-wdom 8464

This theorem is referenced by:  wdomd  8486  brwdom3  8487  unwdomg  8489  xpwdomg  8490  wdom2d2  37602