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Theorem xpdom1g 8001
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 7905 . . . 4 Rel ≼
21brrelexi 5118 . . 3 (𝐴𝐵𝐴 ∈ V)
3 xpcomeng 7996 . . . 4 ((𝐴 ∈ V ∧ 𝐶𝑉) → (𝐴 × 𝐶) ≈ (𝐶 × 𝐴))
43ancoms 469 . . 3 ((𝐶𝑉𝐴 ∈ V) → (𝐴 × 𝐶) ≈ (𝐶 × 𝐴))
52, 4sylan2 491 . 2 ((𝐶𝑉𝐴𝐵) → (𝐴 × 𝐶) ≈ (𝐶 × 𝐴))
6 xpdom2g 8000 . . 3 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
71brrelex2i 5119 . . . 4 (𝐴𝐵𝐵 ∈ V)
8 xpcomeng 7996 . . . 4 ((𝐶𝑉𝐵 ∈ V) → (𝐶 × 𝐵) ≈ (𝐵 × 𝐶))
97, 8sylan2 491 . . 3 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐵) ≈ (𝐵 × 𝐶))
10 domentr 7959 . . 3 (((𝐶 × 𝐴) ≼ (𝐶 × 𝐵) ∧ (𝐶 × 𝐵) ≈ (𝐵 × 𝐶)) → (𝐶 × 𝐴) ≼ (𝐵 × 𝐶))
116, 9, 10syl2anc 692 . 2 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐵 × 𝐶))
12 endomtr 7958 . 2 (((𝐴 × 𝐶) ≈ (𝐶 × 𝐴) ∧ (𝐶 × 𝐴) ≼ (𝐵 × 𝐶)) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))
135, 11, 12syl2anc 692 1 ((𝐶𝑉𝐴𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  Vcvv 3186   class class class wbr 4613   × cxp 5072  cen 7896  cdom 7897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-1st 7113  df-2nd 7114  df-en 7900  df-dom 7901
This theorem is referenced by:  xpdom1  8003  xpen  8067  xpct  8783  infpwfien  8829  fnct  9303  iunctb  9340  canthp1lem1  9418  gchxpidm  9435
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