Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpdom2 Structured version   Visualization version   GIF version

Theorem xpdom2 8015
 Description: Dominance law for Cartesian product. Proposition 10.33(2) of [TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypothesis
Ref Expression
xpdom.2 𝐶 ∈ V
Assertion
Ref Expression
xpdom2 (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))

Proof of Theorem xpdom2
Dummy variables 𝑢 𝑓 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brdomi 7926 . 2 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
2 f1f 6068 . . . . . . . 8 (𝑓:𝐴1-1𝐵𝑓:𝐴𝐵)
3 ffvelrn 6323 . . . . . . . . 9 ((𝑓:𝐴𝐵 ran {𝑥} ∈ 𝐴) → (𝑓 ran {𝑥}) ∈ 𝐵)
43ex 450 . . . . . . . 8 (𝑓:𝐴𝐵 → ( ran {𝑥} ∈ 𝐴 → (𝑓 ran {𝑥}) ∈ 𝐵))
52, 4syl 17 . . . . . . 7 (𝑓:𝐴1-1𝐵 → ( ran {𝑥} ∈ 𝐴 → (𝑓 ran {𝑥}) ∈ 𝐵))
65anim2d 588 . . . . . 6 (𝑓:𝐴1-1𝐵 → (( dom {𝑥} ∈ 𝐶 ran {𝑥} ∈ 𝐴) → ( dom {𝑥} ∈ 𝐶 ∧ (𝑓 ran {𝑥}) ∈ 𝐵)))
76adantld 483 . . . . 5 (𝑓:𝐴1-1𝐵 → ((𝑥 = ⟨ dom {𝑥}, ran {𝑥}⟩ ∧ ( dom {𝑥} ∈ 𝐶 ran {𝑥} ∈ 𝐴)) → ( dom {𝑥} ∈ 𝐶 ∧ (𝑓 ran {𝑥}) ∈ 𝐵)))
8 elxp4 7072 . . . . 5 (𝑥 ∈ (𝐶 × 𝐴) ↔ (𝑥 = ⟨ dom {𝑥}, ran {𝑥}⟩ ∧ ( dom {𝑥} ∈ 𝐶 ran {𝑥} ∈ 𝐴)))
9 opelxp 5116 . . . . 5 (⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ ∈ (𝐶 × 𝐵) ↔ ( dom {𝑥} ∈ 𝐶 ∧ (𝑓 ran {𝑥}) ∈ 𝐵))
107, 8, 93imtr4g 285 . . . 4 (𝑓:𝐴1-1𝐵 → (𝑥 ∈ (𝐶 × 𝐴) → ⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ ∈ (𝐶 × 𝐵)))
1110adantl 482 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝑥 ∈ (𝐶 × 𝐴) → ⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ ∈ (𝐶 × 𝐵)))
12 elxp2 5102 . . . . . 6 (𝑥 ∈ (𝐶 × 𝐴) ↔ ∃𝑧𝐶𝑤𝐴 𝑥 = ⟨𝑧, 𝑤⟩)
13 elxp2 5102 . . . . . 6 (𝑦 ∈ (𝐶 × 𝐴) ↔ ∃𝑣𝐶𝑢𝐴 𝑦 = ⟨𝑣, 𝑢⟩)
14 vex 3193 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
15 fvex 6168 . . . . . . . . . . . . . . . . . 18 (𝑓𝑤) ∈ V
1614, 15opth 4915 . . . . . . . . . . . . . . . . 17 (⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩ ↔ (𝑧 = 𝑣 ∧ (𝑓𝑤) = (𝑓𝑢)))
17 f1fveq 6484 . . . . . . . . . . . . . . . . . . 19 ((𝑓:𝐴1-1𝐵 ∧ (𝑤𝐴𝑢𝐴)) → ((𝑓𝑤) = (𝑓𝑢) ↔ 𝑤 = 𝑢))
1817ancoms 469 . . . . . . . . . . . . . . . . . 18 (((𝑤𝐴𝑢𝐴) ∧ 𝑓:𝐴1-1𝐵) → ((𝑓𝑤) = (𝑓𝑢) ↔ 𝑤 = 𝑢))
1918anbi2d 739 . . . . . . . . . . . . . . . . 17 (((𝑤𝐴𝑢𝐴) ∧ 𝑓:𝐴1-1𝐵) → ((𝑧 = 𝑣 ∧ (𝑓𝑤) = (𝑓𝑢)) ↔ (𝑧 = 𝑣𝑤 = 𝑢)))
2016, 19syl5bb 272 . . . . . . . . . . . . . . . 16 (((𝑤𝐴𝑢𝐴) ∧ 𝑓:𝐴1-1𝐵) → (⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩ ↔ (𝑧 = 𝑣𝑤 = 𝑢)))
2120ex 450 . . . . . . . . . . . . . . 15 ((𝑤𝐴𝑢𝐴) → (𝑓:𝐴1-1𝐵 → (⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩ ↔ (𝑧 = 𝑣𝑤 = 𝑢))))
2221ad2ant2l 781 . . . . . . . . . . . . . 14 (((𝑧𝐶𝑤𝐴) ∧ (𝑣𝐶𝑢𝐴)) → (𝑓:𝐴1-1𝐵 → (⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩ ↔ (𝑧 = 𝑣𝑤 = 𝑢))))
2322imp 445 . . . . . . . . . . . . 13 ((((𝑧𝐶𝑤𝐴) ∧ (𝑣𝐶𝑢𝐴)) ∧ 𝑓:𝐴1-1𝐵) → (⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩ ↔ (𝑧 = 𝑣𝑤 = 𝑢)))
2423adantlr 750 . . . . . . . . . . . 12 (((((𝑧𝐶𝑤𝐴) ∧ (𝑣𝐶𝑢𝐴)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ 𝑓:𝐴1-1𝐵) → (⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩ ↔ (𝑧 = 𝑣𝑤 = 𝑢)))
25 sneq 4165 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑧, 𝑤⟩ → {𝑥} = {⟨𝑧, 𝑤⟩})
2625dmeqd 5296 . . . . . . . . . . . . . . . . 17 (𝑥 = ⟨𝑧, 𝑤⟩ → dom {𝑥} = dom {⟨𝑧, 𝑤⟩})
2726unieqd 4419 . . . . . . . . . . . . . . . 16 (𝑥 = ⟨𝑧, 𝑤⟩ → dom {𝑥} = dom {⟨𝑧, 𝑤⟩})
28 vex 3193 . . . . . . . . . . . . . . . . 17 𝑤 ∈ V
2914, 28op1sta 5586 . . . . . . . . . . . . . . . 16 dom {⟨𝑧, 𝑤⟩} = 𝑧
3027, 29syl6eq 2671 . . . . . . . . . . . . . . 15 (𝑥 = ⟨𝑧, 𝑤⟩ → dom {𝑥} = 𝑧)
3125rneqd 5323 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑧, 𝑤⟩ → ran {𝑥} = ran {⟨𝑧, 𝑤⟩})
3231unieqd 4419 . . . . . . . . . . . . . . . . 17 (𝑥 = ⟨𝑧, 𝑤⟩ → ran {𝑥} = ran {⟨𝑧, 𝑤⟩})
3314, 28op2nda 5589 . . . . . . . . . . . . . . . . 17 ran {⟨𝑧, 𝑤⟩} = 𝑤
3432, 33syl6eq 2671 . . . . . . . . . . . . . . . 16 (𝑥 = ⟨𝑧, 𝑤⟩ → ran {𝑥} = 𝑤)
3534fveq2d 6162 . . . . . . . . . . . . . . 15 (𝑥 = ⟨𝑧, 𝑤⟩ → (𝑓 ran {𝑥}) = (𝑓𝑤))
3630, 35opeq12d 4385 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑧, 𝑤⟩ → ⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨𝑧, (𝑓𝑤)⟩)
37 sneq 4165 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑣, 𝑢⟩ → {𝑦} = {⟨𝑣, 𝑢⟩})
3837dmeqd 5296 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑣, 𝑢⟩ → dom {𝑦} = dom {⟨𝑣, 𝑢⟩})
3938unieqd 4419 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑣, 𝑢⟩ → dom {𝑦} = dom {⟨𝑣, 𝑢⟩})
40 vex 3193 . . . . . . . . . . . . . . . . 17 𝑣 ∈ V
41 vex 3193 . . . . . . . . . . . . . . . . 17 𝑢 ∈ V
4240, 41op1sta 5586 . . . . . . . . . . . . . . . 16 dom {⟨𝑣, 𝑢⟩} = 𝑣
4339, 42syl6eq 2671 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑣, 𝑢⟩ → dom {𝑦} = 𝑣)
4437rneqd 5323 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑣, 𝑢⟩ → ran {𝑦} = ran {⟨𝑣, 𝑢⟩})
4544unieqd 4419 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑣, 𝑢⟩ → ran {𝑦} = ran {⟨𝑣, 𝑢⟩})
4640, 41op2nda 5589 . . . . . . . . . . . . . . . . 17 ran {⟨𝑣, 𝑢⟩} = 𝑢
4745, 46syl6eq 2671 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑣, 𝑢⟩ → ran {𝑦} = 𝑢)
4847fveq2d 6162 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑓 ran {𝑦}) = (𝑓𝑢))
4943, 48opeq12d 4385 . . . . . . . . . . . . . 14 (𝑦 = ⟨𝑣, 𝑢⟩ → ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ = ⟨𝑣, (𝑓𝑢)⟩)
5036, 49eqeqan12d 2637 . . . . . . . . . . . . 13 ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → (⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ ↔ ⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩))
5150ad2antlr 762 . . . . . . . . . . . 12 (((((𝑧𝐶𝑤𝐴) ∧ (𝑣𝐶𝑢𝐴)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ 𝑓:𝐴1-1𝐵) → (⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ ↔ ⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩))
52 eqeq12 2634 . . . . . . . . . . . . . 14 ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → (𝑥 = 𝑦 ↔ ⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩))
5314, 28opth 4915 . . . . . . . . . . . . . 14 (⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩ ↔ (𝑧 = 𝑣𝑤 = 𝑢))
5452, 53syl6bb 276 . . . . . . . . . . . . 13 ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → (𝑥 = 𝑦 ↔ (𝑧 = 𝑣𝑤 = 𝑢)))
5554ad2antlr 762 . . . . . . . . . . . 12 (((((𝑧𝐶𝑤𝐴) ∧ (𝑣𝐶𝑢𝐴)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ 𝑓:𝐴1-1𝐵) → (𝑥 = 𝑦 ↔ (𝑧 = 𝑣𝑤 = 𝑢)))
5624, 51, 553bitr4d 300 . . . . . . . . . . 11 (((((𝑧𝐶𝑤𝐴) ∧ (𝑣𝐶𝑢𝐴)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ 𝑓:𝐴1-1𝐵) → (⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ ↔ 𝑥 = 𝑦))
5756exp53 646 . . . . . . . . . 10 ((𝑧𝐶𝑤𝐴) → ((𝑣𝐶𝑢𝐴) → (𝑥 = ⟨𝑧, 𝑤⟩ → (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑓:𝐴1-1𝐵 → (⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ ↔ 𝑥 = 𝑦))))))
5857com23 86 . . . . . . . . 9 ((𝑧𝐶𝑤𝐴) → (𝑥 = ⟨𝑧, 𝑤⟩ → ((𝑣𝐶𝑢𝐴) → (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑓:𝐴1-1𝐵 → (⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ ↔ 𝑥 = 𝑦))))))
5958rexlimivv 3031 . . . . . . . 8 (∃𝑧𝐶𝑤𝐴 𝑥 = ⟨𝑧, 𝑤⟩ → ((𝑣𝐶𝑢𝐴) → (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑓:𝐴1-1𝐵 → (⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ ↔ 𝑥 = 𝑦)))))
6059rexlimdvv 3032 . . . . . . 7 (∃𝑧𝐶𝑤𝐴 𝑥 = ⟨𝑧, 𝑤⟩ → (∃𝑣𝐶𝑢𝐴 𝑦 = ⟨𝑣, 𝑢⟩ → (𝑓:𝐴1-1𝐵 → (⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ ↔ 𝑥 = 𝑦))))
6160imp 445 . . . . . 6 ((∃𝑧𝐶𝑤𝐴 𝑥 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑣𝐶𝑢𝐴 𝑦 = ⟨𝑣, 𝑢⟩) → (𝑓:𝐴1-1𝐵 → (⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ ↔ 𝑥 = 𝑦)))
6212, 13, 61syl2anb 496 . . . . 5 ((𝑥 ∈ (𝐶 × 𝐴) ∧ 𝑦 ∈ (𝐶 × 𝐴)) → (𝑓:𝐴1-1𝐵 → (⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ ↔ 𝑥 = 𝑦)))
6362com12 32 . . . 4 (𝑓:𝐴1-1𝐵 → ((𝑥 ∈ (𝐶 × 𝐴) ∧ 𝑦 ∈ (𝐶 × 𝐴)) → (⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ ↔ 𝑥 = 𝑦)))
6463adantl 482 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ((𝑥 ∈ (𝐶 × 𝐴) ∧ 𝑦 ∈ (𝐶 × 𝐴)) → (⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ ↔ 𝑥 = 𝑦)))
65 xpdom.2 . . . . 5 𝐶 ∈ V
66 reldom 7921 . . . . . 6 Rel ≼
6766brrelexi 5128 . . . . 5 (𝐴𝐵𝐴 ∈ V)
68 xpexg 6925 . . . . 5 ((𝐶 ∈ V ∧ 𝐴 ∈ V) → (𝐶 × 𝐴) ∈ V)
6965, 67, 68sylancr 694 . . . 4 (𝐴𝐵 → (𝐶 × 𝐴) ∈ V)
7069adantr 481 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐶 × 𝐴) ∈ V)
7166brrelex2i 5129 . . . . 5 (𝐴𝐵𝐵 ∈ V)
72 xpexg 6925 . . . . 5 ((𝐶 ∈ V ∧ 𝐵 ∈ V) → (𝐶 × 𝐵) ∈ V)
7365, 71, 72sylancr 694 . . . 4 (𝐴𝐵 → (𝐶 × 𝐵) ∈ V)
7473adantr 481 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐶 × 𝐵) ∈ V)
7511, 64, 70, 74dom3d 7957 . 2 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
761, 75exlimddv 1860 1 (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∃wrex 2909  Vcvv 3190  {csn 4155  ⟨cop 4161  ∪ cuni 4409   class class class wbr 4623   × cxp 5082  dom cdm 5084  ran crn 5085  ⟶wf 5853  –1-1→wf1 5854  ‘cfv 5857   ≼ cdom 7913 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fv 5865  df-dom 7917 This theorem is referenced by:  xpdom2g  8016  infxpenlem  8796  xpct  8799  cfpwsdom  9366  inar1  9557  rexpen  14901  2ndcctbss  21198  tx1stc  21393  tx2ndc  21394  met2ndci  22267  mbfimaopnlem  23362
 Copyright terms: Public domain W3C validator