Theorem wdom2d 9516

Description: Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 5242). (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 5288	. . . . . 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	3, 4	xpexd 7685	. . . 4 ⊢ (𝜑 → ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} × 𝐴) ∈ V)
6		csbeq1 3858	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ⦋𝑧 / 𝑦⦌𝑋 = ⦋𝑤 / 𝑦⦌𝑋)
7	6	eleq1d 2822	. . . . . . . . 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 497	. . . . . . 7 ⊢ (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} → ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴)
10	9	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}) → ⦋𝑤 / 𝑦⦌𝑋 ∈ 𝐴)
11	10	fmpttd 7063	. . . . 5 ⊢ (𝜑 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋):{𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}⟶𝐴)
12		fssxp 6696	. . . . 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 17	. . . 4 ⊢ (𝜑 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋) ⊆ ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} × 𝐴))
14	5, 13	ssexd 5281	. . 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 2825	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
17	16	biimpcd 248	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑋 → 𝑋 ∈ 𝐴))
18	17	ancrd 552	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑋 → (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)))
19	18	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝑋 → (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)))
20	19	reximdv 3167	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝑥 = 𝑋 → ∃𝑦 ∈ 𝐵 (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)))
21	15, 20	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋))
22		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑣(𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋)
23		nfcsb1v 3880	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝑋
24	23	nfel1 2923	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴
25	23	nfeq2 2924	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 = ⦋𝑣 / 𝑦⦌𝑋
26	24, 25	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑦(⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋)
27		csbeq1a 3869	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → 𝑋 = ⦋𝑣 / 𝑦⦌𝑋)
28	27	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑋 ∈ 𝐴 ↔ ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴))
29	27	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑥 = 𝑋 ↔ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
30	28, 29	anbi12d 631	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) ↔ (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋)))
31	22, 26, 30	cbvrexw 3290	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 (𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) ↔ ∃𝑣 ∈ 𝐵 (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
32	21, 31	sylib 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑣 ∈ 𝐵 (⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴 ∧ 𝑥 = ⦋𝑣 / 𝑦⦌𝑋))
33		csbeq1 3858	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑣 → ⦋𝑧 / 𝑦⦌𝑋 = ⦋𝑣 / 𝑦⦌𝑋)
34	33	eleq1d 2822	. . . . . . . . . . . 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 497	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} → ⦋𝑣 / 𝑦⦌𝑋 ∈ 𝐴)
37		csbeq1 3858	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → ⦋𝑤 / 𝑦⦌𝑋 = ⦋𝑣 / 𝑦⦌𝑋)
38		eqid 2736	. . . . . . . . . . 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 6946	. . . . . . . . . 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 685	. . . . . . . . 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 2747	. . . . . . . 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 3095	. . . . . . 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 274	. . . . . 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 233	. . . . 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 3143	. . . 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 7052	. . . 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 583	. . 3 ⊢ (𝜑 → (𝑤 ∈ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ↦ ⦋𝑤 / 𝑦⦌𝑋):{𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴}–onto→𝐴)
49		fowdom 9507	. . 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 584	. 2 ⊢ (𝜑 → 𝐴 ≼* {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴})
51		ssrab2 4037	. . . 4 ⊢ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ⊆ 𝐵
52		ssdomg 8940	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ⊆ 𝐵 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼ 𝐵))
53	51, 52	mpi 20	. . 3 ⊢ (𝐵 ∈ 𝑊 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼ 𝐵)
54		domwdom 9510	. . 3 ⊢ ({𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼ 𝐵 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼* 𝐵)
55	1, 53, 54	3syl 18	. 2 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼* 𝐵)
56		wdomtr 9511	. 2 ⊢ ((𝐴 ≼* {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ∧ {𝑧 ∈ 𝐵 ∣ ⦋𝑧 / 𝑦⦌𝑋 ∈ 𝐴} ≼* 𝐵) → 𝐴 ≼* 𝐵)
57	50, 55, 56	syl2anc 584	1 ⊢ (𝜑 → 𝐴 ≼* 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445  ⦋csb 3855   ⊆ wss 3910   class class class wbr 5105   ↦ cmpt 5188   × cxp 5631  ⟶wf 6492  –onto→wfo 6494  ‘cfv 6496   ≼ cdom 8881   ≼^* cwdom 9500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-en 8884  df-dom 8885  df-sdom 8886  df-wdom 9501

This theorem is referenced by:  wdomd  9517  brwdom3  9518  unwdomg  9520  xpwdomg  9521  wdom2d2  41345