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Theorem xpdom2g 8298
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 5326 . . . . 5 (𝑥 = 𝐶 → (𝑥 × 𝐴) = (𝐶 × 𝐴))
2 xpeq1 5326 . . . . 5 (𝑥 = 𝐶 → (𝑥 × 𝐵) = (𝐶 × 𝐵))
31, 2breq12d 4856 . . . 4 (𝑥 = 𝐶 → ((𝑥 × 𝐴) ≼ (𝑥 × 𝐵) ↔ (𝐶 × 𝐴) ≼ (𝐶 × 𝐵)))
43imbi2d 332 . . 3 (𝑥 = 𝐶 → ((𝐴𝐵 → (𝑥 × 𝐴) ≼ (𝑥 × 𝐵)) ↔ (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))))
5 vex 3388 . . . 4 𝑥 ∈ V
65xpdom2 8297 . . 3 (𝐴𝐵 → (𝑥 × 𝐴) ≼ (𝑥 × 𝐵))
74, 6vtoclg 3453 . 2 (𝐶𝑉 → (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵)))
87imp 396 1 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157   class class class wbr 4843   × cxp 5310  cdom 8193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fv 6109  df-dom 8197
This theorem is referenced by:  xpdom1g  8299  xpen  8365  infcdaabs  9316  infxpdom  9321  fin56  9503  fnct  9647  unirnfdomd  9677  pwcdandom  9777  gchxpidm  9779  gchhar  9789
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