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Theorem xpdom2g 8607
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 5568 . . . . 5 (𝑥 = 𝐶 → (𝑥 × 𝐴) = (𝐶 × 𝐴))
2 xpeq1 5568 . . . . 5 (𝑥 = 𝐶 → (𝑥 × 𝐵) = (𝐶 × 𝐵))
31, 2breq12d 5076 . . . 4 (𝑥 = 𝐶 → ((𝑥 × 𝐴) ≼ (𝑥 × 𝐵) ↔ (𝐶 × 𝐴) ≼ (𝐶 × 𝐵)))
43imbi2d 342 . . 3 (𝑥 = 𝐶 → ((𝐴𝐵 → (𝑥 × 𝐴) ≼ (𝑥 × 𝐵)) ↔ (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))))
5 vex 3503 . . . 4 𝑥 ∈ V
65xpdom2 8606 . . 3 (𝐴𝐵 → (𝑥 × 𝐴) ≼ (𝑥 × 𝐵))
74, 6vtoclg 3573 . 2 (𝐶𝑉 → (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵)))
87imp 407 1 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107   class class class wbr 5063   × cxp 5552  cdom 8501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fv 6362  df-dom 8505
This theorem is referenced by:  xpdom1g  8608  xpen  8674  infxpdom  9627  fnct  9953  unirnfdomd  9983  gchxpidm  10085  gchhar  10095
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