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Theorem xpdom1g 8098
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 8003 . . . 4 Rel ≼
21brrelexi 5192 . . 3 (𝐴𝐵𝐴 ∈ V)
3 xpcomeng 8093 . . . 4 ((𝐴 ∈ V ∧ 𝐶𝑉) → (𝐴 × 𝐶) ≈ (𝐶 × 𝐴))
43ancoms 468 . . 3 ((𝐶𝑉𝐴 ∈ V) → (𝐴 × 𝐶) ≈ (𝐶 × 𝐴))
52, 4sylan2 490 . 2 ((𝐶𝑉𝐴𝐵) → (𝐴 × 𝐶) ≈ (𝐶 × 𝐴))
6 xpdom2g 8097 . . 3 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
71brrelex2i 5193 . . . 4 (𝐴𝐵𝐵 ∈ V)
8 xpcomeng 8093 . . . 4 ((𝐶𝑉𝐵 ∈ V) → (𝐶 × 𝐵) ≈ (𝐵 × 𝐶))
97, 8sylan2 490 . . 3 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐵) ≈ (𝐵 × 𝐶))
10 domentr 8056 . . 3 (((𝐶 × 𝐴) ≼ (𝐶 × 𝐵) ∧ (𝐶 × 𝐵) ≈ (𝐵 × 𝐶)) → (𝐶 × 𝐴) ≼ (𝐵 × 𝐶))
116, 9, 10syl2anc 694 . 2 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐵 × 𝐶))
12 endomtr 8055 . 2 (((𝐴 × 𝐶) ≈ (𝐶 × 𝐴) ∧ (𝐶 × 𝐴) ≼ (𝐵 × 𝐶)) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))
135, 11, 12syl2anc 694 1 ((𝐶𝑉𝐴𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  Vcvv 3231   class class class wbr 4685   × cxp 5141  cen 7994  cdom 7995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1st 7210  df-2nd 7211  df-en 7998  df-dom 7999
This theorem is referenced by:  xpdom1  8100  xpen  8164  xpct  8877  infpwfien  8923  fnct  9397  iunctb  9434  canthp1lem1  9512  gchxpidm  9529
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