Theorem xpwdomg 9582

Description: Weak dominance of a Cartesian product. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)

Assertion

Ref	Expression
xpwdomg	⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))

Proof of Theorem xpwdomg

Dummy variables 𝑎 𝑏 𝑐 𝑓 𝑔 𝑥 𝑦 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brwdom3i 9580	. . 3 ⊢ (𝐴 ≼* 𝐵 → ∃𝑓∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏))
2	1	adantr 479	. 2 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → ∃𝑓∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏))
3		brwdom3i 9580	. . 3 ⊢ (𝐶 ≼* 𝐷 → ∃𝑔∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))
4	3	adantl 480	. 2 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → ∃𝑔∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))
5		relwdom 9563	. . . . . . . . . 10 ⊢ Rel ≼*
6	5	brrelex1i 5731	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐴 ∈ V)
7	5	brrelex1i 5731	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐶 ∈ V)
8		xpexg 7739	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 × 𝐶) ∈ V)
9	6, 7, 8	syl2an 594	. . . . . . . 8 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐴 × 𝐶) ∈ V)
10	9	adantr 479	. . . . . . 7 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) ∧ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))) → (𝐴 × 𝐶) ∈ V)
11	5	brrelex2i 5732	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐵 ∈ V)
12	5	brrelex2i 5732	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐷 ∈ V)
13		xpexg 7739	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 × 𝐷) ∈ V)
14	11, 12, 13	syl2an 594	. . . . . . . 8 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐵 × 𝐷) ∈ V)
15	14	adantr 479	. . . . . . 7 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) ∧ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))) → (𝐵 × 𝐷) ∈ V)
16		pm3.2 468	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) → (∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑) → (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))))
17	16	ralimdv 3167	. . . . . . . . . . . . . . 15 ⊢ (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) → (∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑) → ∀𝑐 ∈ 𝐶 (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))))
18	17	com12 32	. . . . . . . . . . . . . 14 ⊢ (∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑) → (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) → ∀𝑐 ∈ 𝐶 (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))))
19	18	ralimdv 3167	. . . . . . . . . . . . 13 ⊢ (∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑) → (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) → ∀𝑎 ∈ 𝐴 ∀𝑐 ∈ 𝐶 (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))))
20	19	impcom 406	. . . . . . . . . . . 12 ⊢ ((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑)) → ∀𝑎 ∈ 𝐴 ∀𝑐 ∈ 𝐶 (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑)))
21		pm3.2 468	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑓‘𝑏) → (𝑐 = (𝑔‘𝑑) → (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑))))
22	21	reximdv 3168	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (𝑓‘𝑏) → (∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑) → ∃𝑑 ∈ 𝐷 (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑))))
23	22	com12 32	. . . . . . . . . . . . . . 15 ⊢ (∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑) → (𝑎 = (𝑓‘𝑏) → ∃𝑑 ∈ 𝐷 (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑))))
24	23	reximdv 3168	. . . . . . . . . . . . . 14 ⊢ (∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑) → (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) → ∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑))))
25	24	impcom 406	. . . . . . . . . . . . 13 ⊢ ((∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑)) → ∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑)))
26	25	2ralimi 3121	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑐 ∈ 𝐶 (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑)) → ∀𝑎 ∈ 𝐴 ∀𝑐 ∈ 𝐶 ∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑)))
27	20, 26	syl 17	. . . . . . . . . . 11 ⊢ ((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑)) → ∀𝑎 ∈ 𝐴 ∀𝑐 ∈ 𝐶 ∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑)))
28		eqeq1 2734	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑎, 𝑐⟩ → (𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩ ↔ ⟨𝑎, 𝑐⟩ = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩))
29		vex 3476	. . . . . . . . . . . . . . 15 ⊢ 𝑎 ∈ V
30		vex 3476	. . . . . . . . . . . . . . 15 ⊢ 𝑐 ∈ V
31	29, 30	opth 5475	. . . . . . . . . . . . . 14 ⊢ (⟨𝑎, 𝑐⟩ = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩ ↔ (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑)))
32	28, 31	bitrdi 286	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑎, 𝑐⟩ → (𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩ ↔ (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑))))
33	32	2rexbidv 3217	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑎, 𝑐⟩ → (∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩ ↔ ∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑))))
34	33	ralxp 5840	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝐴 × 𝐶)∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩ ↔ ∀𝑎 ∈ 𝐴 ∀𝑐 ∈ 𝐶 ∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑)))
35	27, 34	sylibr 233	. . . . . . . . . 10 ⊢ ((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑)) → ∀𝑥 ∈ (𝐴 × 𝐶)∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩)
36	35	r19.21bi 3246	. . . . . . . . 9 ⊢ (((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑)) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩)
37		vex 3476	. . . . . . . . . . . . . 14 ⊢ 𝑏 ∈ V
38		vex 3476	. . . . . . . . . . . . . 14 ⊢ 𝑑 ∈ V
39	37, 38	op1std 7987	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (1st ‘𝑦) = 𝑏)
40	39	fveq2d 6894	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑓‘(1st ‘𝑦)) = (𝑓‘𝑏))
41	37, 38	op2ndd 7988	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (2nd ‘𝑦) = 𝑑)
42	41	fveq2d 6894	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑔‘(2nd ‘𝑦)) = (𝑔‘𝑑))
43	40, 42	opeq12d 4880	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → ⟨(𝑓‘(1st ‘𝑦)), (𝑔‘(2nd ‘𝑦))⟩ = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩)
44	43	eqeq2d 2741	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑥 = ⟨(𝑓‘(1st ‘𝑦)), (𝑔‘(2nd ‘𝑦))⟩ ↔ 𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩))
45	44	rexxp 5841	. . . . . . . . 9 ⊢ (∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st ‘𝑦)), (𝑔‘(2nd ‘𝑦))⟩ ↔ ∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩)
46	36, 45	sylibr 233	. . . . . . . 8 ⊢ (((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑)) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st ‘𝑦)), (𝑔‘(2nd ‘𝑦))⟩)
47	46	adantll 710	. . . . . . 7 ⊢ ((((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) ∧ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st ‘𝑦)), (𝑔‘(2nd ‘𝑦))⟩)
48	10, 15, 47	wdom2d 9577	. . . . . 6 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) ∧ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))
49	48	expr 455	. . . . 5 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → (∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷)))
50	49	exlimdv 1934	. . . 4 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → (∃𝑔∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷)))
51	50	ex 411	. . 3 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) → (∃𝑔∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))))
52	51	exlimdv 1934	. 2 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (∃𝑓∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) → (∃𝑔∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))))
53	2, 4, 52	mp2d 49	1 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539  ∃wex 1779   ∈ wcel 2104  ∀wral 3059  ∃wrex 3068  Vcvv 3472  ⟨cop 4633   class class class wbr 5147   × cxp 5673  ‘cfv 6542  1^st c1st 7975  2^nd c2nd 7976   ≼^* cwdom 9561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1st 7977  df-2nd 7978  df-en 8942  df-dom 8943  df-sdom 8944  df-wdom 9562

This theorem is referenced by:  hsmexlem3  10425