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Theorem xpdom2g 9019
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 5652 . . . . 5 (𝑥 = 𝐶 → (𝑥 × 𝐴) = (𝐶 × 𝐴))
2 xpeq1 5652 . . . . 5 (𝑥 = 𝐶 → (𝑥 × 𝐵) = (𝐶 × 𝐵))
31, 2breq12d 5123 . . . 4 (𝑥 = 𝐶 → ((𝑥 × 𝐴) ≼ (𝑥 × 𝐵) ↔ (𝐶 × 𝐴) ≼ (𝐶 × 𝐵)))
43imbi2d 341 . . 3 (𝑥 = 𝐶 → ((𝐴𝐵 → (𝑥 × 𝐴) ≼ (𝑥 × 𝐵)) ↔ (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))))
5 vex 3452 . . . 4 𝑥 ∈ V
65xpdom2 9018 . . 3 (𝐴𝐵 → (𝑥 × 𝐴) ≼ (𝑥 × 𝐵))
74, 6vtoclg 3528 . 2 (𝐶𝑉 → (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵)))
87imp 408 1 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107   class class class wbr 5110   × cxp 5636  cdom 8888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fv 6509  df-dom 8892
This theorem is referenced by:  xpdom1g  9020  xpen  9091  infxpdom  10154  fnct  10480  unirnfdomd  10510  gchxpidm  10612  gchhar  10622
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