MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpwdomg Structured version   Visualization version   GIF version

Theorem xpwdomg 9579
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 9577 . . 3 (𝐴* 𝐵 → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
21adantr 481 . 2 ((𝐴* 𝐵𝐶* 𝐷) → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
3 brwdom3i 9577 . . 3 (𝐶* 𝐷 → ∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))
43adantl 482 . 2 ((𝐴* 𝐵𝐶* 𝐷) → ∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))
5 relwdom 9560 . . . . . . . . . 10 Rel ≼*
65brrelex1i 5732 . . . . . . . . 9 (𝐴* 𝐵𝐴 ∈ V)
75brrelex1i 5732 . . . . . . . . 9 (𝐶* 𝐷𝐶 ∈ V)
8 xpexg 7736 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 × 𝐶) ∈ V)
96, 7, 8syl2an 596 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐴 × 𝐶) ∈ V)
109adantr 481 . . . . . . 7 (((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) → (𝐴 × 𝐶) ∈ V)
115brrelex2i 5733 . . . . . . . . 9 (𝐴* 𝐵𝐵 ∈ V)
125brrelex2i 5733 . . . . . . . . 9 (𝐶* 𝐷𝐷 ∈ V)
13 xpexg 7736 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 × 𝐷) ∈ V)
1411, 12, 13syl2an 596 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐵 × 𝐷) ∈ V)
1514adantr 481 . . . . . . 7 (((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) → (𝐵 × 𝐷) ∈ V)
16 pm3.2 470 . . . . . . . . . . . . . . . 16 (∃𝑏𝐵 𝑎 = (𝑓𝑏) → (∃𝑑𝐷 𝑐 = (𝑔𝑑) → (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑))))
1716ralimdv 3169 . . . . . . . . . . . . . . 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 3169 . . . . . . . . . . . . 13 (∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) → ∀𝑎𝐴𝑐𝐶 (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑))))
2019impcom 408 . . . . . . . . . . . 12 ((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) → ∀𝑎𝐴𝑐𝐶 (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑)))
21 pm3.2 470 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝑓𝑏) → (𝑐 = (𝑔𝑑) → (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
2221reximdv 3170 . . . . . . . . . . . . . . . 16 (𝑎 = (𝑓𝑏) → (∃𝑑𝐷 𝑐 = (𝑔𝑑) → ∃𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
2322com12 32 . . . . . . . . . . . . . . 15 (∃𝑑𝐷 𝑐 = (𝑔𝑑) → (𝑎 = (𝑓𝑏) → ∃𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
2423reximdv 3170 . . . . . . . . . . . . . 14 (∃𝑑𝐷 𝑐 = (𝑔𝑑) → (∃𝑏𝐵 𝑎 = (𝑓𝑏) → ∃𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
2524impcom 408 . . . . . . . . . . . . 13 ((∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑)) → ∃𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
26252ralimi 3123 . . . . . . . . . . . 12 (∀𝑎𝐴𝑐𝐶 (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑)) → ∀𝑎𝐴𝑐𝐶𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
2720, 26syl 17 . . . . . . . . . . 11 ((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) → ∀𝑎𝐴𝑐𝐶𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
28 eqeq1 2736 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑎, 𝑐⟩ → (𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ ⟨𝑎, 𝑐⟩ = ⟨(𝑓𝑏), (𝑔𝑑)⟩))
29 vex 3478 . . . . . . . . . . . . . . 15 𝑎 ∈ V
30 vex 3478 . . . . . . . . . . . . . . 15 𝑐 ∈ V
3129, 30opth 5476 . . . . . . . . . . . . . 14 (⟨𝑎, 𝑐⟩ = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
3228, 31bitrdi 286 . . . . . . . . . . . . 13 (𝑥 = ⟨𝑎, 𝑐⟩ → (𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
33322rexbidv 3219 . . . . . . . . . . . 12 (𝑥 = ⟨𝑎, 𝑐⟩ → (∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ ∃𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
3433ralxp 5841 . . . . . . . . . . 11 (∀𝑥 ∈ (𝐴 × 𝐶)∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ ∀𝑎𝐴𝑐𝐶𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
3527, 34sylibr 233 . . . . . . . . . 10 ((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) → ∀𝑥 ∈ (𝐴 × 𝐶)∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
3635r19.21bi 3248 . . . . . . . . 9 (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
37 vex 3478 . . . . . . . . . . . . . 14 𝑏 ∈ V
38 vex 3478 . . . . . . . . . . . . . 14 𝑑 ∈ V
3937, 38op1std 7984 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑏, 𝑑⟩ → (1st𝑦) = 𝑏)
4039fveq2d 6895 . . . . . . . . . . . 12 (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑓‘(1st𝑦)) = (𝑓𝑏))
4137, 38op2ndd 7985 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑏, 𝑑⟩ → (2nd𝑦) = 𝑑)
4241fveq2d 6895 . . . . . . . . . . . 12 (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑔‘(2nd𝑦)) = (𝑔𝑑))
4340, 42opeq12d 4881 . . . . . . . . . . 11 (𝑦 = ⟨𝑏, 𝑑⟩ → ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩ = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
4443eqeq2d 2743 . . . . . . . . . 10 (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩ ↔ 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩))
4544rexxp 5842 . . . . . . . . 9 (∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩ ↔ ∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
4636, 45sylibr 233 . . . . . . . 8 (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩)
4746adantll 712 . . . . . . 7 ((((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩)
4810, 15, 47wdom2d 9574 . . . . . 6 (((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))
4948expr 457 . . . . 5 (((𝐴* 𝐵𝐶* 𝐷) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → (∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷)))
5049exlimdv 1936 . . . 4 (((𝐴* 𝐵𝐶* 𝐷) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → (∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷)))
5150ex 413 . . 3 ((𝐴* 𝐵𝐶* 𝐷) → (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) → (∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))))
5251exlimdv 1936 . 2 ((𝐴* 𝐵𝐶* 𝐷) → (∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) → (∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))))
532, 4, 52mp2d 49 1 ((𝐴* 𝐵𝐶* 𝐷) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wex 1781  wcel 2106  wral 3061  wrex 3070  Vcvv 3474  cop 4634   class class class wbr 5148   × cxp 5674  cfv 6543  1st c1st 7972  2nd c2nd 7973  * cwdom 9558
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1st 7974  df-2nd 7975  df-en 8939  df-dom 8940  df-sdom 8941  df-wdom 9559
This theorem is referenced by:  hsmexlem3  10422
  Copyright terms: Public domain W3C validator