Theorem xpwdomg 9625

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 9623	. . 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 9623	. . 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 9606	. . . . . . . . . 10 ⊢ Rel ≼*
6	5	brrelex1i 5741	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐴 ∈ V)
7	5	brrelex1i 5741	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐶 ∈ V)
8		xpexg 7770	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 × 𝐶) ∈ V)
9	6, 7, 8	syl2an 596	. . . . . . . 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 5742	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐵 ∈ V)
12	5	brrelex2i 5742	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐷 ∈ V)
13		xpexg 7770	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 × 𝐷) ∈ V)
14	11, 12, 13	syl2an 596	. . . . . . . 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 3169	. . . . . . . . . . . . . . 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 3169	. . . . . . . . . . . . 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 3170	. . . . . . . . . . . . . . . 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 3170	. . . . . . . . . . . . . 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 3123	. . . . . . . . . . . 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 2741	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑎, 𝑐⟩ → (𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩ ↔ ⟨𝑎, 𝑐⟩ = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩))
29		vex 3484	. . . . . . . . . . . . . . 15 ⊢ 𝑎 ∈ V
30		vex 3484	. . . . . . . . . . . . . . 15 ⊢ 𝑐 ∈ V
31	29, 30	opth 5481	. . . . . . . . . . . . . 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 3222	. . . . . . . . . . . 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 5852	. . . . . . . . . . 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 3251	. . . . . . . . 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 3484	. . . . . . . . . . . . . 14 ⊢ 𝑏 ∈ V
38		vex 3484	. . . . . . . . . . . . . 14 ⊢ 𝑑 ∈ V
39	37, 38	op1std 8024	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (1st ‘𝑦) = 𝑏)
40	39	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑓‘(1st ‘𝑦)) = (𝑓‘𝑏))
41	37, 38	op2ndd 8025	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (2nd ‘𝑦) = 𝑑)
42	41	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑔‘(2nd ‘𝑦)) = (𝑔‘𝑑))
43	40, 42	opeq12d 4881	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → ⟨(𝑓‘(1st ‘𝑦)), (𝑔‘(2nd ‘𝑦))⟩ = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩)
44	43	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑥 = ⟨(𝑓‘(1st ‘𝑦)), (𝑔‘(2nd ‘𝑦))⟩ ↔ 𝑥 = ⟨(𝑓‘𝑏), (𝑔‘𝑑)⟩))
45	44	rexxp 5853	. . . . . . . . 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 714	. . . . . . 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 9620	. . . . . 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 1933	. . . 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 1933	. 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 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  Vcvv 3480  ⟨cop 4632   class class class wbr 5143   × cxp 5683  ‘cfv 6561  1^st c1st 8012  2^nd c2nd 8013   ≼^* cwdom 9604

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015  df-en 8986  df-dom 8987  df-sdom 8988  df-wdom 9605

This theorem is referenced by:  hsmexlem3  10468