Theorem xpwdomg 9500

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 9498	. . 3 ⊢ (𝐴 ≼* 𝐵 → ∃𝑓∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏))
2	1	adantr 480	. 2 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → ∃𝑓∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏))
3		brwdom3i 9498	. . 3 ⊢ (𝐶 ≼* 𝐷 → ∃𝑔∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))
4	3	adantl 481	. 2 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → ∃𝑔∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))
5		relwdom 9481	. . . . . . . . . 10 ⊢ Rel ≼*
6	5	brrelex1i 5687	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐴 ∈ V)
7	5	brrelex1i 5687	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐶 ∈ V)
8		xpexg 7704	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 × 𝐶) ∈ V)
9	6, 7, 8	syl2an 597	. . . . . . . 8 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐴 × 𝐶) ∈ V)
10	9	adantr 480	. . . . . . 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 5688	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐵 ∈ V)
12	5	brrelex2i 5688	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐷 ∈ V)
13		xpexg 7704	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 × 𝐷) ∈ V)
14	11, 12, 13	syl2an 597	. . . . . . . 8 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐵 × 𝐷) ∈ V)
15	14	adantr 480	. . . . . . 7 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) ∧ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))) → (𝐵 × 𝐷) ∈ V)
16		pm3.2 469	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) → (∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑) → (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑))))
17	16	ralimdv 3151	. . . . . . . . . . . . . . 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 3151	. . . . . . . . . . . . 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 407	. . . . . . . . . . . 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		pm3.2 469	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑓‘𝑏) → (𝑐 = (𝑔‘𝑑) → (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑))))
22	21	reximdv 3152	. . . . . . . . . . . . . . . 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 3152	. . . . . . . . . . . . . 14 ⊢ (∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑) → (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) → ∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑))))
25	24	impcom 407	. . . . . . . . . . . . 13 ⊢ ((∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∃𝑑 ∈ 𝐷 𝑐 = (𝑔‘𝑑)) → ∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑)))
26	25	2ralimi 3107	. . . . . . . . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑎, 𝑐⟩ → (𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩ ↔ ⟨𝑎, 𝑐⟩ = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩))
29		vex 3433	. . . . . . . . . . . . . . 15 ⊢ 𝑎 ∈ V
30		vex 3433	. . . . . . . . . . . . . . 15 ⊢ 𝑐 ∈ V
31	29, 30	opth 5429	. . . . . . . . . . . . . 14 ⊢ (⟨𝑎, 𝑐⟩ = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩ ↔ (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑)))
32	28, 31	bitrdi 287	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑎, 𝑐⟩ → (𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩ ↔ (𝑎 = (𝑓‘𝑏) ∧ 𝑐 = (𝑔‘𝑑))))
33	32	2rexbidv 3202	. . . . . . . . . . . 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 5796	. . . . . . . . . . 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 234	. . . . . . . . . 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 3229	. . . . . . . . 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 3433	. . . . . . . . . . . . . 14 ⊢ 𝑏 ∈ V
38		vex 3433	. . . . . . . . . . . . . 14 ⊢ 𝑑 ∈ V
39	37, 38	op1std 7952	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (1st ‘𝑦) = 𝑏)
40	39	fveq2d 6844	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑓‘(1st ‘𝑦)) = (𝑓‘𝑏))
41	37, 38	op2ndd 7953	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (2nd ‘𝑦) = 𝑑)
42	41	fveq2d 6844	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑔‘(2nd ‘𝑦)) = (𝑔‘𝑑))
43	40, 42	opeq12d 4824	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → ⟨(𝑓‘(1st ‘𝑦)), (𝑔‘(2nd ‘𝑦))⟩ = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩)
44	43	eqeq2d 2747	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑥 = ⟨(𝑓‘(1st ‘𝑦)), (𝑔‘(2nd ‘𝑦))⟩ ↔ 𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩))
45	44	rexxp 5797	. . . . . . . . 9 ⊢ (∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st ‘𝑦)), (𝑔‘(2nd ‘𝑦))⟩ ↔ ∃𝑏 ∈ 𝐵 ∃𝑑 ∈ 𝐷 𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩)
46	36, 45	sylibr 234	. . . . . . . 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 715	. . . . . . 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 9495	. . . . . 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 456	. . . . 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 1935	. . . 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 412	. . 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 1935	. 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 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  Vcvv 3429  ⟨cop 4573   class class class wbr 5085   × cxp 5629  ‘cfv 6498  1^st c1st 7940  2^nd c2nd 7941   ≼^* cwdom 9479

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-1st 7942  df-2nd 7943  df-en 8894  df-dom 8895  df-sdom 8896  df-wdom 9480

This theorem is referenced by:  hsmexlem3  10350