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Theorem xpdom2 8096
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 8008 . 2 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
2 f1f 6139 . . . . . . . 8 (𝑓:𝐴1-1𝐵𝑓:𝐴𝐵)
3 ffvelrn 6397 . . . . . . . . 9 ((𝑓:𝐴𝐵 ran {𝑥} ∈ 𝐴) → (𝑓 ran {𝑥}) ∈ 𝐵)
43ex 449 . . . . . . . 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 482 . . . . 5 (𝑓:𝐴1-1𝐵 → ((𝑥 = ⟨ dom {𝑥}, ran {𝑥}⟩ ∧ ( dom {𝑥} ∈ 𝐶 ran {𝑥} ∈ 𝐴)) → ( dom {𝑥} ∈ 𝐶 ∧ (𝑓 ran {𝑥}) ∈ 𝐵)))
8 elxp4 7152 . . . . 5 (𝑥 ∈ (𝐶 × 𝐴) ↔ (𝑥 = ⟨ dom {𝑥}, ran {𝑥}⟩ ∧ ( dom {𝑥} ∈ 𝐶 ran {𝑥} ∈ 𝐴)))
9 opelxp 5180 . . . . 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 481 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝑥 ∈ (𝐶 × 𝐴) → ⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ ∈ (𝐶 × 𝐵)))
12 elxp2 5166 . . . . . 6 (𝑥 ∈ (𝐶 × 𝐴) ↔ ∃𝑧𝐶𝑤𝐴 𝑥 = ⟨𝑧, 𝑤⟩)
13 elxp2 5166 . . . . . 6 (𝑦 ∈ (𝐶 × 𝐴) ↔ ∃𝑣𝐶𝑢𝐴 𝑦 = ⟨𝑣, 𝑢⟩)
14 vex 3234 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
15 fvex 6239 . . . . . . . . . . . . . . . . . 18 (𝑓𝑤) ∈ V
1614, 15opth 4974 . . . . . . . . . . . . . . . . 17 (⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩ ↔ (𝑧 = 𝑣 ∧ (𝑓𝑤) = (𝑓𝑢)))
17 f1fveq 6559 . . . . . . . . . . . . . . . . . . 19 ((𝑓:𝐴1-1𝐵 ∧ (𝑤𝐴𝑢𝐴)) → ((𝑓𝑤) = (𝑓𝑢) ↔ 𝑤 = 𝑢))
1817ancoms 468 . . . . . . . . . . . . . . . . . 18 (((𝑤𝐴𝑢𝐴) ∧ 𝑓:𝐴1-1𝐵) → ((𝑓𝑤) = (𝑓𝑢) ↔ 𝑤 = 𝑢))
1918anbi2d 740 . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . 15 ((𝑤𝐴𝑢𝐴) → (𝑓:𝐴1-1𝐵 → (⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩ ↔ (𝑧 = 𝑣𝑤 = 𝑢))))
2221ad2ant2l 797 . . . . . . . . . . . . . 14 (((𝑧𝐶𝑤𝐴) ∧ (𝑣𝐶𝑢𝐴)) → (𝑓:𝐴1-1𝐵 → (⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩ ↔ (𝑧 = 𝑣𝑤 = 𝑢))))
2322imp 444 . . . . . . . . . . . . 13 ((((𝑧𝐶𝑤𝐴) ∧ (𝑣𝐶𝑢𝐴)) ∧ 𝑓:𝐴1-1𝐵) → (⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩ ↔ (𝑧 = 𝑣𝑤 = 𝑢)))
2423adantlr 751 . . . . . . . . . . . 12 (((((𝑧𝐶𝑤𝐴) ∧ (𝑣𝐶𝑢𝐴)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ 𝑓:𝐴1-1𝐵) → (⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩ ↔ (𝑧 = 𝑣𝑤 = 𝑢)))
25 sneq 4220 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑧, 𝑤⟩ → {𝑥} = {⟨𝑧, 𝑤⟩})
2625dmeqd 5358 . . . . . . . . . . . . . . . . 17 (𝑥 = ⟨𝑧, 𝑤⟩ → dom {𝑥} = dom {⟨𝑧, 𝑤⟩})
2726unieqd 4478 . . . . . . . . . . . . . . . 16 (𝑥 = ⟨𝑧, 𝑤⟩ → dom {𝑥} = dom {⟨𝑧, 𝑤⟩})
28 vex 3234 . . . . . . . . . . . . . . . . 17 𝑤 ∈ V
2914, 28op1sta 5654 . . . . . . . . . . . . . . . 16 dom {⟨𝑧, 𝑤⟩} = 𝑧
3027, 29syl6eq 2701 . . . . . . . . . . . . . . 15 (𝑥 = ⟨𝑧, 𝑤⟩ → dom {𝑥} = 𝑧)
3125rneqd 5385 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑧, 𝑤⟩ → ran {𝑥} = ran {⟨𝑧, 𝑤⟩})
3231unieqd 4478 . . . . . . . . . . . . . . . . 17 (𝑥 = ⟨𝑧, 𝑤⟩ → ran {𝑥} = ran {⟨𝑧, 𝑤⟩})
3314, 28op2nda 5658 . . . . . . . . . . . . . . . . 17 ran {⟨𝑧, 𝑤⟩} = 𝑤
3432, 33syl6eq 2701 . . . . . . . . . . . . . . . 16 (𝑥 = ⟨𝑧, 𝑤⟩ → ran {𝑥} = 𝑤)
3534fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑥 = ⟨𝑧, 𝑤⟩ → (𝑓 ran {𝑥}) = (𝑓𝑤))
3630, 35opeq12d 4441 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑧, 𝑤⟩ → ⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨𝑧, (𝑓𝑤)⟩)
37 sneq 4220 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑣, 𝑢⟩ → {𝑦} = {⟨𝑣, 𝑢⟩})
3837dmeqd 5358 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑣, 𝑢⟩ → dom {𝑦} = dom {⟨𝑣, 𝑢⟩})
3938unieqd 4478 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑣, 𝑢⟩ → dom {𝑦} = dom {⟨𝑣, 𝑢⟩})
40 vex 3234 . . . . . . . . . . . . . . . . 17 𝑣 ∈ V
41 vex 3234 . . . . . . . . . . . . . . . . 17 𝑢 ∈ V
4240, 41op1sta 5654 . . . . . . . . . . . . . . . 16 dom {⟨𝑣, 𝑢⟩} = 𝑣
4339, 42syl6eq 2701 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑣, 𝑢⟩ → dom {𝑦} = 𝑣)
4437rneqd 5385 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑣, 𝑢⟩ → ran {𝑦} = ran {⟨𝑣, 𝑢⟩})
4544unieqd 4478 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑣, 𝑢⟩ → ran {𝑦} = ran {⟨𝑣, 𝑢⟩})
4640, 41op2nda 5658 . . . . . . . . . . . . . . . . 17 ran {⟨𝑣, 𝑢⟩} = 𝑢
4745, 46syl6eq 2701 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑣, 𝑢⟩ → ran {𝑦} = 𝑢)
4847fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑓 ran {𝑦}) = (𝑓𝑢))
4943, 48opeq12d 4441 . . . . . . . . . . . . . 14 (𝑦 = ⟨𝑣, 𝑢⟩ → ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ = ⟨𝑣, (𝑓𝑢)⟩)
5036, 49eqeqan12d 2667 . . . . . . . . . . . . 13 ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → (⟨ dom {𝑥}, (𝑓 ran {𝑥})⟩ = ⟨ dom {𝑦}, (𝑓 ran {𝑦})⟩ ↔ ⟨𝑧, (𝑓𝑤)⟩ = ⟨𝑣, (𝑓𝑢)⟩))
5150ad2antlr 763 . . . . . . . . . . . 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 2664 . . . . . . . . . . . . . 14 ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → (𝑥 = 𝑦 ↔ ⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩))
5314, 28opth 4974 . . . . . . . . . . . . . 14 (⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩ ↔ (𝑧 = 𝑣𝑤 = 𝑢))
5452, 53syl6bb 276 . . . . . . . . . . . . 13 ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → (𝑥 = 𝑦 ↔ (𝑧 = 𝑣𝑤 = 𝑢)))
5554ad2antlr 763 . . . . . . . . . . . 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 3065 . . . . . . . 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 3066 . . . . . . 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 444 . . . . . 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 495 . . . . 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 481 . . 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 8003 . . . . . 6 Rel ≼
6766brrelexi 5192 . . . . 5 (𝐴𝐵𝐴 ∈ V)
68 xpexg 7002 . . . . 5 ((𝐶 ∈ V ∧ 𝐴 ∈ V) → (𝐶 × 𝐴) ∈ V)
6965, 67, 68sylancr 696 . . . 4 (𝐴𝐵 → (𝐶 × 𝐴) ∈ V)
7069adantr 480 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐶 × 𝐴) ∈ V)
7166brrelex2i 5193 . . . . 5 (𝐴𝐵𝐵 ∈ V)
72 xpexg 7002 . . . . 5 ((𝐶 ∈ V ∧ 𝐵 ∈ V) → (𝐶 × 𝐵) ∈ V)
7365, 71, 72sylancr 696 . . . 4 (𝐴𝐵 → (𝐶 × 𝐵) ∈ V)
7473adantr 480 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐶 × 𝐵) ∈ V)
7511, 64, 70, 74dom3d 8039 . 2 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
761, 75exlimddv 1903 1 (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231  {csn 4210  cop 4216   cuni 4468   class class class wbr 4685   × cxp 5141  dom cdm 5143  ran crn 5144  wf 5922  1-1wf1 5923  cfv 5926  cdom 7995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fv 5934  df-dom 7999
This theorem is referenced by:  xpdom2g  8097  infxpenlem  8874  xpct  8877  cfpwsdom  9444  inar1  9635  rexpen  15001  2ndcctbss  21306  tx1stc  21501  tx2ndc  21502  met2ndci  22374  mbfimaopnlem  23467
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