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Theorem xpdom2g 9107
Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
xpdom2g ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))

Proof of Theorem xpdom2g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 xpeq1 5703 . . . . 5 (𝑥 = 𝐶 → (𝑥 × 𝐴) = (𝐶 × 𝐴))
2 xpeq1 5703 . . . . 5 (𝑥 = 𝐶 → (𝑥 × 𝐵) = (𝐶 × 𝐵))
31, 2breq12d 5161 . . . 4 (𝑥 = 𝐶 → ((𝑥 × 𝐴) ≼ (𝑥 × 𝐵) ↔ (𝐶 × 𝐴) ≼ (𝐶 × 𝐵)))
43imbi2d 340 . . 3 (𝑥 = 𝐶 → ((𝐴𝐵 → (𝑥 × 𝐴) ≼ (𝑥 × 𝐵)) ↔ (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))))
5 vex 3482 . . . 4 𝑥 ∈ V
65xpdom2 9106 . . 3 (𝐴𝐵 → (𝑥 × 𝐴) ≼ (𝑥 × 𝐵))
74, 6vtoclg 3554 . 2 (𝐶𝑉 → (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵)))
87imp 406 1 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106   class class class wbr 5148   × cxp 5687  cdom 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571  df-dom 8986
This theorem is referenced by:  xpdom1g  9108  xpen  9179  infxpdom  10248  fnct  10575  unirnfdomd  10605  gchxpidm  10707  gchhar  10717
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