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Theorem xpdom2g 6242
Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
xpdom2g ((𝐶 𝑉 AB) → (𝐶 × A) ≼ (𝐶 × B))

Proof of Theorem xpdom2g
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 xpeq1 4302 . . . . 5 (x = 𝐶 → (x × A) = (𝐶 × A))
2 xpeq1 4302 . . . . 5 (x = 𝐶 → (x × B) = (𝐶 × B))
31, 2breq12d 3768 . . . 4 (x = 𝐶 → ((x × A) ≼ (x × B) ↔ (𝐶 × A) ≼ (𝐶 × B)))
43imbi2d 219 . . 3 (x = 𝐶 → ((AB → (x × A) ≼ (x × B)) ↔ (AB → (𝐶 × A) ≼ (𝐶 × B))))
5 vex 2554 . . . 4 x V
65xpdom2 6241 . . 3 (AB → (x × A) ≼ (x × B))
74, 6vtoclg 2607 . 2 (𝐶 𝑉 → (AB → (𝐶 × A) ≼ (𝐶 × B)))
87imp 115 1 ((𝐶 𝑉 AB) → (𝐶 × A) ≼ (𝐶 × B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390   class class class wbr 3755   × cxp 4286  cdom 6156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fv 4853  df-dom 6159
This theorem is referenced by:  xpdom1g  6243
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