ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpdom2g GIF version

Theorem xpdom2g 6502
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 4427 . . . . 5 (𝑥 = 𝐶 → (𝑥 × 𝐴) = (𝐶 × 𝐴))
2 xpeq1 4427 . . . . 5 (𝑥 = 𝐶 → (𝑥 × 𝐵) = (𝐶 × 𝐵))
31, 2breq12d 3835 . . . 4 (𝑥 = 𝐶 → ((𝑥 × 𝐴) ≼ (𝑥 × 𝐵) ↔ (𝐶 × 𝐴) ≼ (𝐶 × 𝐵)))
43imbi2d 228 . . 3 (𝑥 = 𝐶 → ((𝐴𝐵 → (𝑥 × 𝐴) ≼ (𝑥 × 𝐵)) ↔ (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))))
5 vex 2618 . . . 4 𝑥 ∈ V
65xpdom2 6501 . . 3 (𝐴𝐵 → (𝑥 × 𝐴) ≼ (𝑥 × 𝐵))
74, 6vtoclg 2672 . 2 (𝐶𝑉 → (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵)))
87imp 122 1 ((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wcel 1436   class class class wbr 3822   × cxp 4411  cdom 6410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-mpt 3878  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fv 4991  df-dom 6413
This theorem is referenced by:  xpdom1g  6503
  Copyright terms: Public domain W3C validator