Theorem wdom2d 9567

Description: Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 5232). (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 5299	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ∈ V)
3	1, 2	syl 18	. . . . 5 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ∈ V)
4		wdom2d.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5	3, 4	xpexd 7763	. . . 4 ⊢ (𝜑 → ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} × 𝐴) ∈ V)
6		csbeq1 3850	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ⦋𝑧 / 𝑦⦌𝑋 = ⦋𝑤 / 𝑦⦌𝑋)
7	6	eleq1d 2846	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴 ↔ ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴))
8	7	elrab 3645	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↔ (𝑤 ∈ 𝐵 ∧ ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴))
9	8	simprbi 503	. . . . . . 7 ⊢ (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} → ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴)
10	9	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}) → ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴)
11	10	fmpttd 7113	. . . . 5 ⊢ (𝜑 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋):{𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}⟶𝐴)
12		fssxp 6735	. . . . 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))
13	11, 12	syl 18	. . . 4 ⊢ (𝜑 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋) ⊆ ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} × 𝐴))
14	5, 13	ssexd 5286	. . 3 ⊢ (𝜑 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋) ∈ V)
15		wdom2d.o	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)
16		eleq1 2849	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
17	16	biimpcd 252	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑋 → 𝑋 ∈ 𝐴))
18	17	ancrd 561	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑋 → (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)))
19	18	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝑋 → (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)))
20	19	reximdv 3178	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝑥 = 𝑋 → ∃𝑦 ∈ 𝐵 (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)))
21	15, 20	mpd 16	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋))
22		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑣(𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)
23		nfcsb1v 3871	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝑋
24	23	nfel1 2939	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴
25	23	nfeq2 2940	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 = ⦋𝑣 / 𝑦⦌𝑋
26	24, 25	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑦(⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋)
27		csbeq1a 3861	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → 𝑋 = ⦋𝑣 / 𝑦⦌𝑋)
28	27	eleq1d 2846	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑋 ∈ 𝐴 ↔ ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴))
29	27	eqeq2d 2772	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑥 = 𝑋 ↔ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
30	28, 29	anbi12d 644	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) ↔ (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋)))
31	22, 26, 30	cbvrexw 3306	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) ↔ ∃𝑣 ∈ 𝐵 (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
32	21, 31	sylib 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑣 ∈ 𝐵 (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
33		csbeq1 3850	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑣 → ⦋𝑧 / 𝑦⦌𝑋 = ⦋𝑣 / 𝑦⦌𝑋)
34	33	eleq1d 2846	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑣 → (⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴 ↔ ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴))
35	34	elrab 3645	. . . . . . . . . . 11 ⊢ (𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↔ (𝑣 ∈ 𝐵 ∧ ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴))
36	35	simprbi 503	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} → ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴)
37		csbeq1 3850	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → ⦋𝑤 / 𝑦⦌𝑋 = ⦋𝑣 / 𝑦⦌𝑋)
38		eqid 2761	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋) = (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)
39	37, 38	fvmptg 6989	. . . . . . . . . 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)
40	36, 39	mpdan 700	. . . . . . . . 9 ⊢ (𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} → ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)‘𝑣) = ⦋𝑣 / 𝑦⦌𝑋)
41	40	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} → (𝑥 = ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)‘𝑣) ↔ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
42	41	rexbiia 3108	. . . . . . 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)
43	34	rexrab 3654	. . . . . . 7 ⊢ (∃𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}𝑥 = ⦋𝑣 / 𝑦⦌𝑋 ↔ ∃𝑣 ∈ 𝐵 (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
44	42, 43	bitri 278	. . . . . 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))
45	32, 44	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}𝑥 = ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)‘𝑣))
46	45	ralrimiva 3155	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}𝑥 = ((𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋)‘𝑣))
47		dffo3 7100	. . . 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)))
48	11, 46, 47	sylanbrc 595	. . 3 ⊢ (𝜑 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋):{𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}–onto→𝐴)
49		fowdom 9558	. . 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})
50	14, 48, 49	syl2anc 596	. 2 ⊢ (𝜑 → 𝐴 ≼* {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴})
51		ssrab2 4028	. . . 4 ⊢ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ⊆ 𝐵
52		ssdomg 9020	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ⊆ 𝐵 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼ 𝐵))
53	51, 52	mpi 21	. . 3 ⊢ (𝐵 ∈ 𝑊 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼ 𝐵)
54		domwdom 9561	. . 3 ⊢ ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼ 𝐵 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼* 𝐵)
55	1, 53, 54	3syl 19	. 2 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼* 𝐵)
56		wdomtr 9562	. 2 ⊢ ((𝐴 ≼* {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ∧ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼* 𝐵) → 𝐴 ≼* 𝐵)
57	50, 55, 56	syl2anc 596	1 ⊢ (𝜑 → 𝐴 ≼* 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451  ⦋csb 3847   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186   × cxp 5649  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537   ≼ cdom 8964   ≼^* cwdom 9551

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-en 8967  df-dom 8968  df-sdom 8969  df-wdom 9552

This theorem is used by:  wdomd  9568  brwdom3  9569  unwdomg  9571  xpwdomg  9572  wdom2d2  44021