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Theorem xpwdomg 9037
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 9035 . . 3 (𝐴* 𝐵 → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
21adantr 481 . 2 ((𝐴* 𝐵𝐶* 𝐷) → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
3 brwdom3i 9035 . . 3 (𝐶* 𝐷 → ∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))
43adantl 482 . 2 ((𝐴* 𝐵𝐶* 𝐷) → ∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))
5 relwdom 9018 . . . . . . . . . 10 Rel ≼*
65brrelex1i 5601 . . . . . . . . 9 (𝐴* 𝐵𝐴 ∈ V)
75brrelex1i 5601 . . . . . . . . 9 (𝐶* 𝐷𝐶 ∈ V)
8 xpexg 7462 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 × 𝐶) ∈ V)
96, 7, 8syl2an 595 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐴 × 𝐶) ∈ V)
109adantr 481 . . . . . . 7 (((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) → (𝐴 × 𝐶) ∈ V)
115brrelex2i 5602 . . . . . . . . 9 (𝐴* 𝐵𝐵 ∈ V)
125brrelex2i 5602 . . . . . . . . 9 (𝐶* 𝐷𝐷 ∈ V)
13 xpexg 7462 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 × 𝐷) ∈ V)
1411, 12, 13syl2an 595 . . . . . . . 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 3175 . . . . . . . . . . . . . . 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 3175 . . . . . . . . . . . . 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 3270 . . . . . . . . . . . . . . . 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 3270 . . . . . . . . . . . . . 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 3158 . . . . . . . . . . . 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 2822 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑎, 𝑐⟩ → (𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ ⟨𝑎, 𝑐⟩ = ⟨(𝑓𝑏), (𝑔𝑑)⟩))
29 vex 3495 . . . . . . . . . . . . . . 15 𝑎 ∈ V
30 vex 3495 . . . . . . . . . . . . . . 15 𝑐 ∈ V
3129, 30opth 5359 . . . . . . . . . . . . . 14 (⟨𝑎, 𝑐⟩ = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
3228, 31syl6bb 288 . . . . . . . . . . . . 13 (𝑥 = ⟨𝑎, 𝑐⟩ → (𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
33322rexbidv 3297 . . . . . . . . . . . 12 (𝑥 = ⟨𝑎, 𝑐⟩ → (∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ ∃𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
3433ralxp 5705 . . . . . . . . . . 11 (∀𝑥 ∈ (𝐴 × 𝐶)∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ ∀𝑎𝐴𝑐𝐶𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
3527, 34sylibr 235 . . . . . . . . . 10 ((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) → ∀𝑥 ∈ (𝐴 × 𝐶)∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
3635r19.21bi 3205 . . . . . . . . 9 (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
37 vex 3495 . . . . . . . . . . . . . 14 𝑏 ∈ V
38 vex 3495 . . . . . . . . . . . . . 14 𝑑 ∈ V
3937, 38op1std 7688 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑏, 𝑑⟩ → (1st𝑦) = 𝑏)
4039fveq2d 6667 . . . . . . . . . . . 12 (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑓‘(1st𝑦)) = (𝑓𝑏))
4137, 38op2ndd 7689 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑏, 𝑑⟩ → (2nd𝑦) = 𝑑)
4241fveq2d 6667 . . . . . . . . . . . 12 (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑔‘(2nd𝑦)) = (𝑔𝑑))
4340, 42opeq12d 4803 . . . . . . . . . . 11 (𝑦 = ⟨𝑏, 𝑑⟩ → ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩ = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
4443eqeq2d 2829 . . . . . . . . . 10 (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩ ↔ 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩))
4544rexxp 5706 . . . . . . . . 9 (∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩ ↔ ∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
4636, 45sylibr 235 . . . . . . . 8 (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩)
4746adantll 710 . . . . . . 7 ((((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩)
4810, 15, 47wdom2d 9032 . . . . . 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 1925 . . . 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 1925 . 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 1528  wex 1771  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  cop 4563   class class class wbr 5057   × cxp 5546  cfv 6348  1st c1st 7676  2nd c2nd 7677  * cwdom 9009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-1st 7678  df-2nd 7679  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-wdom 9011
This theorem is referenced by:  hsmexlem3  9838
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