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Theorem xpdom1g 8602
Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
xpdom1g ((𝐶𝑉𝐴𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))

Proof of Theorem xpdom1g
StepHypRef Expression
1 reldom 8503 . . . 4 Rel ≼
21brrelex1i 5601 . . 3 (𝐴𝐵𝐴 ∈ V)
3 xpcomeng 8597 . . . 4 ((𝐴 ∈ V ∧ 𝐶𝑉) → (𝐴 × 𝐶) ≈ (𝐶 × 𝐴))
43ancoms 459 . . 3 ((𝐶𝑉𝐴 ∈ V) → (𝐴 × 𝐶) ≈ (𝐶 × 𝐴))
52, 4sylan2 592 . 2 ((𝐶𝑉𝐴𝐵) → (𝐴 × 𝐶) ≈ (𝐶 × 𝐴))
6 xpdom2g 8601 . . 3 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
71brrelex2i 5602 . . . 4 (𝐴𝐵𝐵 ∈ V)
8 xpcomeng 8597 . . . 4 ((𝐶𝑉𝐵 ∈ V) → (𝐶 × 𝐵) ≈ (𝐵 × 𝐶))
97, 8sylan2 592 . . 3 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐵) ≈ (𝐵 × 𝐶))
10 domentr 8556 . . 3 (((𝐶 × 𝐴) ≼ (𝐶 × 𝐵) ∧ (𝐶 × 𝐵) ≈ (𝐵 × 𝐶)) → (𝐶 × 𝐴) ≼ (𝐵 × 𝐶))
116, 9, 10syl2anc 584 . 2 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐵 × 𝐶))
12 endomtr 8555 . 2 (((𝐴 × 𝐶) ≈ (𝐶 × 𝐴) ∧ (𝐶 × 𝐴) ≼ (𝐵 × 𝐶)) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))
135, 11, 12syl2anc 584 1 ((𝐶𝑉𝐴𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  Vcvv 3492   class class class wbr 5057   × cxp 5546  cen 8494  cdom 8495
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-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-en 8498  df-dom 8499
This theorem is referenced by:  xpdom1  8604  xpen  8668  xpct  9430  infpwfien  9476  infdjuabs  9616  fin56  9803  fnct  9947  iunctb  9984  canthp1lem1  10062  pwdjundom  10077  gchxpidm  10079
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