Theorem wdom2d 8368

Description: Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 4699). (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 4739	. . . . . 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 6858	. . . . 5 ⊢ (({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ∈ V ∧ 𝐴 ∈ 𝑉) → ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} × 𝐴) ∈ V)
6	3, 4, 5	syl2anc 691	. . . 4 ⊢ (𝜑 → ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} × 𝐴) ∈ V)
7		csbeq1 3502	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ⦋𝑧 / 𝑦⦌𝑋 = ⦋𝑤 / 𝑦⦌𝑋)
8	7	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴 ↔ ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴))
9	8	elrab 3331	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↔ (𝑤 ∈ 𝐵 ∧ ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴))
10	9	simprbi 479	. . . . . . 7 ⊢ (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} → ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴)
11	10	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}) → ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴)
12		eqid 2610	. . . . . 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 6292	. . . . 5 ⊢ (𝜑 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋):{𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}⟶𝐴)
14		fssxp 5973	. . . . 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 4733	. . 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 2676	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
19	18	biimpcd 238	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑋 → 𝑋 ∈ 𝐴))
20	19	ancrd 575	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑋 → (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)))
21	20	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝑋 → (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)))
22	21	reximdv 2999	. . . . . . . 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 1830	. . . . . . . 8 ⊢ Ⅎ𝑣(𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)
25		nfcsb1v 3515	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝑋
26	25	nfel1 2765	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴
27	25	nfeq2 2766	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 = ⦋𝑣 / 𝑦⦌𝑋
28	26, 27	nfan 1816	. . . . . . . 8 ⊢ Ⅎ𝑦(⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋)
29		csbeq1a 3508	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → 𝑋 = ⦋𝑣 / 𝑦⦌𝑋)
30	29	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑋 ∈ 𝐴 ↔ ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴))
31	29	eqeq2d 2620	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑥 = 𝑋 ↔ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
32	30, 31	anbi12d 743	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) ↔ (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋)))
33	24, 28, 32	cbvrex 3144	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) ↔ ∃𝑣 ∈ 𝐵 (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
34	23, 33	sylib 207	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑣 ∈ 𝐵 (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
35		csbeq1 3502	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑣 → ⦋𝑧 / 𝑦⦌𝑋 = ⦋𝑣 / 𝑦⦌𝑋)
36	35	eleq1d 2672	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑣 → (⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴 ↔ ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴))
37	36	elrab 3331	. . . . . . . . . . 11 ⊢ (𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↔ (𝑣 ∈ 𝐵 ∧ ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴))
38	37	simprbi 479	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} → ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴)
39		csbeq1 3502	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → ⦋𝑤 / 𝑦⦌𝑋 = ⦋𝑣 / 𝑦⦌𝑋)
40	39, 12	fvmptg 6189	. . . . . . . . . 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 699	. . . . . . . . 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 2620	. . . . . . . 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 3022	. . . . . . 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 3337	. . . . . . 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 263	. . . . . 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 223	. . . . 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 2949	. . . 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 6282	. . . 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 695	. . 3 ⊢ (𝜑 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋):{𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}–onto→𝐴)
50		fowdom 8359	. . 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 691	. 2 ⊢ (𝜑 → 𝐴 ≼* {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴})
52		ssrab2 3650	. . . 4 ⊢ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ⊆ 𝐵
53		ssdomg 7887	. . . 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 8362	. . 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 8363	. 2 ⊢ ((𝐴 ≼* {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ∧ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼* 𝐵) → 𝐴 ≼* 𝐵)
58	51, 56, 57	syl2anc 691	1 ⊢ (𝜑 → 𝐴 ≼* 𝐵)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ⦋csb 3499   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804   ≼ cdom 7839   ≼^* cwdom 8345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-wdom 8347

This theorem is referenced by:  wdomd  8369  brwdom3  8370  unwdomg  8372  xpwdomg  8373  wdom2d2  36620