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Theorem xpdom1g 6243
Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
xpdom1g ((𝐶 𝑉 AB) → (A × 𝐶) ≼ (B × 𝐶))

Proof of Theorem xpdom1g
StepHypRef Expression
1 reldom 6162 . . . 4 Rel ≼
21brrelexi 4327 . . 3 (ABA V)
3 xpcomeng 6238 . . . 4 ((A V 𝐶 𝑉) → (A × 𝐶) ≈ (𝐶 × A))
43ancoms 255 . . 3 ((𝐶 𝑉 A V) → (A × 𝐶) ≈ (𝐶 × A))
52, 4sylan2 270 . 2 ((𝐶 𝑉 AB) → (A × 𝐶) ≈ (𝐶 × A))
6 xpdom2g 6242 . . 3 ((𝐶 𝑉 AB) → (𝐶 × A) ≼ (𝐶 × B))
71brrelex2i 4328 . . . 4 (ABB V)
8 xpcomeng 6238 . . . 4 ((𝐶 𝑉 B V) → (𝐶 × B) ≈ (B × 𝐶))
97, 8sylan2 270 . . 3 ((𝐶 𝑉 AB) → (𝐶 × B) ≈ (B × 𝐶))
10 domentr 6207 . . 3 (((𝐶 × A) ≼ (𝐶 × B) (𝐶 × B) ≈ (B × 𝐶)) → (𝐶 × A) ≼ (B × 𝐶))
116, 9, 10syl2anc 391 . 2 ((𝐶 𝑉 AB) → (𝐶 × A) ≼ (B × 𝐶))
12 endomtr 6206 . 2 (((A × 𝐶) ≈ (𝐶 × A) (𝐶 × A) ≼ (B × 𝐶)) → (A × 𝐶) ≼ (B × 𝐶))
135, 11, 12syl2anc 391 1 ((𝐶 𝑉 AB) → (A × 𝐶) ≼ (B × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  Vcvv 2551   class class class wbr 3755   × cxp 4286  cen 6155  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-bnd 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-fo 4851  df-f1o 4852  df-fv 4853  df-1st 5709  df-2nd 5710  df-en 6158  df-dom 6159
This theorem is referenced by:  xpdom1  6245
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