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Theorem xpwdomg 8434
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* 𝐵𝐶* 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.
StepHypRef Expression
1 brwdom3i 8432 . . 3 (𝐴* 𝐵 → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
21adantr 481 . 2 ((𝐴* 𝐵𝐶* 𝐷) → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
3 brwdom3i 8432 . . 3 (𝐶* 𝐷 → ∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))
43adantl 482 . 2 ((𝐴* 𝐵𝐶* 𝐷) → ∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))
5 relwdom 8415 . . . . . . . . . 10 Rel ≼*
65brrelexi 5118 . . . . . . . . 9 (𝐴* 𝐵𝐴 ∈ V)
75brrelexi 5118 . . . . . . . . 9 (𝐶* 𝐷𝐶 ∈ V)
8 xpexg 6913 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 × 𝐶) ∈ V)
96, 7, 8syl2an 494 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐴 × 𝐶) ∈ V)
109adantr 481 . . . . . . 7 (((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) → (𝐴 × 𝐶) ∈ V)
115brrelex2i 5119 . . . . . . . . 9 (𝐴* 𝐵𝐵 ∈ V)
125brrelex2i 5119 . . . . . . . . 9 (𝐶* 𝐷𝐷 ∈ V)
13 xpexg 6913 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 × 𝐷) ∈ V)
1411, 12, 13syl2an 494 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐵 × 𝐷) ∈ V)
1514adantr 481 . . . . . . 7 (((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) → (𝐵 × 𝐷) ∈ V)
16 pm3.2 463 . . . . . . . . . . . . . . . 16 (∃𝑏𝐵 𝑎 = (𝑓𝑏) → (∃𝑑𝐷 𝑐 = (𝑔𝑑) → (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑))))
1716ralimdv 2957 . . . . . . . . . . . . . . 15 (∃𝑏𝐵 𝑎 = (𝑓𝑏) → (∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → ∀𝑐𝐶 (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑))))
1817com12 32 . . . . . . . . . . . . . 14 (∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (∃𝑏𝐵 𝑎 = (𝑓𝑏) → ∀𝑐𝐶 (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑))))
1918ralimdv 2957 . . . . . . . . . . . . 13 (∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) → ∀𝑎𝐴𝑐𝐶 (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑))))
2019impcom 446 . . . . . . . . . . . 12 ((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) → ∀𝑎𝐴𝑐𝐶 (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑)))
21 pm3.2 463 . . . . . . . . . . . . . . . . . 18 (𝑎 = (𝑓𝑏) → (𝑐 = (𝑔𝑑) → (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
2221reximdv 3010 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝑓𝑏) → (∃𝑑𝐷 𝑐 = (𝑔𝑑) → ∃𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
2322com12 32 . . . . . . . . . . . . . . . 16 (∃𝑑𝐷 𝑐 = (𝑔𝑑) → (𝑎 = (𝑓𝑏) → ∃𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
2423reximdv 3010 . . . . . . . . . . . . . . 15 (∃𝑑𝐷 𝑐 = (𝑔𝑑) → (∃𝑏𝐵 𝑎 = (𝑓𝑏) → ∃𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
2524impcom 446 . . . . . . . . . . . . . 14 ((∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑)) → ∃𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
2625ralimi 2947 . . . . . . . . . . . . 13 (∀𝑐𝐶 (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑)) → ∀𝑐𝐶𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
2726ralimi 2947 . . . . . . . . . . . 12 (∀𝑎𝐴𝑐𝐶 (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑)) → ∀𝑎𝐴𝑐𝐶𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
2820, 27syl 17 . . . . . . . . . . 11 ((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) → ∀𝑎𝐴𝑐𝐶𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
29 eqeq1 2625 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑎, 𝑐⟩ → (𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ ⟨𝑎, 𝑐⟩ = ⟨(𝑓𝑏), (𝑔𝑑)⟩))
30 vex 3189 . . . . . . . . . . . . . . 15 𝑎 ∈ V
31 vex 3189 . . . . . . . . . . . . . . 15 𝑐 ∈ V
3230, 31opth 4905 . . . . . . . . . . . . . 14 (⟨𝑎, 𝑐⟩ = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
3329, 32syl6bb 276 . . . . . . . . . . . . 13 (𝑥 = ⟨𝑎, 𝑐⟩ → (𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
34332rexbidv 3050 . . . . . . . . . . . 12 (𝑥 = ⟨𝑎, 𝑐⟩ → (∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ ∃𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
3534ralxp 5223 . . . . . . . . . . 11 (∀𝑥 ∈ (𝐴 × 𝐶)∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ ∀𝑎𝐴𝑐𝐶𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
3628, 35sylibr 224 . . . . . . . . . 10 ((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) → ∀𝑥 ∈ (𝐴 × 𝐶)∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
3736r19.21bi 2927 . . . . . . . . 9 (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
38 vex 3189 . . . . . . . . . . . . . 14 𝑏 ∈ V
39 vex 3189 . . . . . . . . . . . . . 14 𝑑 ∈ V
4038, 39op1std 7123 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑏, 𝑑⟩ → (1st𝑦) = 𝑏)
4140fveq2d 6152 . . . . . . . . . . . 12 (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑓‘(1st𝑦)) = (𝑓𝑏))
4238, 39op2ndd 7124 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑏, 𝑑⟩ → (2nd𝑦) = 𝑑)
4342fveq2d 6152 . . . . . . . . . . . 12 (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑔‘(2nd𝑦)) = (𝑔𝑑))
4441, 43opeq12d 4378 . . . . . . . . . . 11 (𝑦 = ⟨𝑏, 𝑑⟩ → ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩ = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
4544eqeq2d 2631 . . . . . . . . . 10 (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩ ↔ 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩))
4645rexxp 5224 . . . . . . . . 9 (∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩ ↔ ∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
4737, 46sylibr 224 . . . . . . . 8 (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩)
4847adantll 749 . . . . . . 7 ((((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩)
4910, 15, 48wdom2d 8429 . . . . . 6 (((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))
5049expr 642 . . . . 5 (((𝐴* 𝐵𝐶* 𝐷) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → (∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷)))
5150exlimdv 1858 . . . 4 (((𝐴* 𝐵𝐶* 𝐷) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → (∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷)))
5251ex 450 . . 3 ((𝐴* 𝐵𝐶* 𝐷) → (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) → (∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))))
5352exlimdv 1858 . 2 ((𝐴* 𝐵𝐶* 𝐷) → (∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) → (∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))))
542, 4, 53mp2d 49 1 ((𝐴* 𝐵𝐶* 𝐷) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cop 4154   class class class wbr 4613   × cxp 5072  cfv 5847  1st c1st 7111  2nd c2nd 7112  * cwdom 8406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-1st 7113  df-2nd 7114  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-wdom 8408
This theorem is referenced by:  hsmexlem3  9194
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