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Theorem xpwdomg 8646
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 8644 . . 3 (𝐴* 𝐵 → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
21adantr 466 . 2 ((𝐴* 𝐵𝐶* 𝐷) → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
3 brwdom3i 8644 . . 3 (𝐶* 𝐷 → ∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))
43adantl 467 . 2 ((𝐴* 𝐵𝐶* 𝐷) → ∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))
5 relwdom 8627 . . . . . . . . . 10 Rel ≼*
65brrelexi 5298 . . . . . . . . 9 (𝐴* 𝐵𝐴 ∈ V)
75brrelexi 5298 . . . . . . . . 9 (𝐶* 𝐷𝐶 ∈ V)
8 xpexg 7107 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 × 𝐶) ∈ V)
96, 7, 8syl2an 583 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐴 × 𝐶) ∈ V)
109adantr 466 . . . . . . 7 (((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) → (𝐴 × 𝐶) ∈ V)
115brrelex2i 5299 . . . . . . . . 9 (𝐴* 𝐵𝐵 ∈ V)
125brrelex2i 5299 . . . . . . . . 9 (𝐶* 𝐷𝐷 ∈ V)
13 xpexg 7107 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 × 𝐷) ∈ V)
1411, 12, 13syl2an 583 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐵 × 𝐷) ∈ V)
1514adantr 466 . . . . . . 7 (((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) → (𝐵 × 𝐷) ∈ V)
16 pm3.2 446 . . . . . . . . . . . . . . . 16 (∃𝑏𝐵 𝑎 = (𝑓𝑏) → (∃𝑑𝐷 𝑐 = (𝑔𝑑) → (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑))))
1716ralimdv 3112 . . . . . . . . . . . . . . 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 3112 . . . . . . . . . . . . 13 (∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) → ∀𝑎𝐴𝑐𝐶 (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑))))
2019impcom 394 . . . . . . . . . . . 12 ((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) → ∀𝑎𝐴𝑐𝐶 (∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑)))
21 pm3.2 446 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝑓𝑏) → (𝑐 = (𝑔𝑑) → (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
2221reximdv 3164 . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . 14 (∃𝑑𝐷 𝑐 = (𝑔𝑑) → (∃𝑏𝐵 𝑎 = (𝑓𝑏) → ∃𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
2524impcom 394 . . . . . . . . . . . . 13 ((∃𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∃𝑑𝐷 𝑐 = (𝑔𝑑)) → ∃𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
26252ralimi 3102 . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑎, 𝑐⟩ → (𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ ⟨𝑎, 𝑐⟩ = ⟨(𝑓𝑏), (𝑔𝑑)⟩))
29 vex 3354 . . . . . . . . . . . . . . 15 𝑎 ∈ V
30 vex 3354 . . . . . . . . . . . . . . 15 𝑐 ∈ V
3129, 30opth 5072 . . . . . . . . . . . . . 14 (⟨𝑎, 𝑐⟩ = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
3228, 31syl6bb 276 . . . . . . . . . . . . 13 (𝑥 = ⟨𝑎, 𝑐⟩ → (𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
33322rexbidv 3205 . . . . . . . . . . . 12 (𝑥 = ⟨𝑎, 𝑐⟩ → (∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ ∃𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑))))
3433ralxp 5402 . . . . . . . . . . 11 (∀𝑥 ∈ (𝐴 × 𝐶)∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩ ↔ ∀𝑎𝐴𝑐𝐶𝑏𝐵𝑑𝐷 (𝑎 = (𝑓𝑏) ∧ 𝑐 = (𝑔𝑑)))
3527, 34sylibr 224 . . . . . . . . . 10 ((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) → ∀𝑥 ∈ (𝐴 × 𝐶)∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
3635r19.21bi 3081 . . . . . . . . 9 (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
37 vex 3354 . . . . . . . . . . . . . 14 𝑏 ∈ V
38 vex 3354 . . . . . . . . . . . . . 14 𝑑 ∈ V
3937, 38op1std 7325 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑏, 𝑑⟩ → (1st𝑦) = 𝑏)
4039fveq2d 6336 . . . . . . . . . . . 12 (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑓‘(1st𝑦)) = (𝑓𝑏))
4137, 38op2ndd 7326 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑏, 𝑑⟩ → (2nd𝑦) = 𝑑)
4241fveq2d 6336 . . . . . . . . . . . 12 (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑔‘(2nd𝑦)) = (𝑔𝑑))
4340, 42opeq12d 4547 . . . . . . . . . . 11 (𝑦 = ⟨𝑏, 𝑑⟩ → ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩ = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
4443eqeq2d 2781 . . . . . . . . . 10 (𝑦 = ⟨𝑏, 𝑑⟩ → (𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩ ↔ 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩))
4544rexxp 5403 . . . . . . . . 9 (∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩ ↔ ∃𝑏𝐵𝑑𝐷 𝑥 = ⟨(𝑓𝑏), (𝑔𝑑)⟩)
4636, 45sylibr 224 . . . . . . . 8 (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑)) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩)
4746adantll 693 . . . . . . 7 ((((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) ∧ 𝑥 ∈ (𝐴 × 𝐶)) → ∃𝑦 ∈ (𝐵 × 𝐷)𝑥 = ⟨(𝑓‘(1st𝑦)), (𝑔‘(2nd𝑦))⟩)
4810, 15, 47wdom2d 8641 . . . . . 6 (((𝐴* 𝐵𝐶* 𝐷) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑))) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))
4948expr 444 . . . . 5 (((𝐴* 𝐵𝐶* 𝐷) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → (∀𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷)))
5049exlimdv 2013 . . . 4 (((𝐴* 𝐵𝐶* 𝐷) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → (∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷)))
5150ex 397 . . 3 ((𝐴* 𝐵𝐶* 𝐷) → (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) → (∃𝑔𝑐𝐶𝑑𝐷 𝑐 = (𝑔𝑑) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))))
5251exlimdv 2013 . 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 382   = wceq 1631  wex 1852  wcel 2145  wral 3061  wrex 3062  Vcvv 3351  cop 4322   class class class wbr 4786   × cxp 5247  cfv 6031  1st c1st 7313  2nd c2nd 7314  * cwdom 8618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-1st 7315  df-2nd 7316  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-wdom 8620
This theorem is referenced by:  hsmexlem3  9452
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