Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  brwdom2 Structured version   Visualization version   GIF version

Theorem brwdom2 8361
 Description: Alternate characterization of the weak dominance predicate which does not require special treatment of the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Assertion
Ref Expression
brwdom2 (𝑌𝑉 → (𝑋* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
Distinct variable groups:   𝑦,𝑋,𝑧   𝑦,𝑌,𝑧
Allowed substitution hints:   𝑉(𝑦,𝑧)

Proof of Theorem brwdom2
Dummy variables 𝑥 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝑌𝑉𝑌 ∈ V)
2 0wdom 8358 . . . . . 6 (𝑌 ∈ V → ∅ ≼* 𝑌)
3 breq1 4586 . . . . . 6 (𝑋 = ∅ → (𝑋* 𝑌 ↔ ∅ ≼* 𝑌))
42, 3syl5ibrcom 236 . . . . 5 (𝑌 ∈ V → (𝑋 = ∅ → 𝑋* 𝑌))
54imp 444 . . . 4 ((𝑌 ∈ V ∧ 𝑋 = ∅) → 𝑋* 𝑌)
6 0elpw 4760 . . . . . . 7 ∅ ∈ 𝒫 𝑌
7 f1o0 6085 . . . . . . . 8 ∅:∅–1-1-onto→∅
8 f1ofo 6057 . . . . . . . 8 (∅:∅–1-1-onto→∅ → ∅:∅–onto→∅)
9 0ex 4718 . . . . . . . . 9 ∅ ∈ V
10 foeq1 6024 . . . . . . . . 9 (𝑧 = ∅ → (𝑧:∅–onto→∅ ↔ ∅:∅–onto→∅))
119, 10spcev 3273 . . . . . . . 8 (∅:∅–onto→∅ → ∃𝑧 𝑧:∅–onto→∅)
127, 8, 11mp2b 10 . . . . . . 7 𝑧 𝑧:∅–onto→∅
13 foeq2 6025 . . . . . . . . 9 (𝑦 = ∅ → (𝑧:𝑦onto→∅ ↔ 𝑧:∅–onto→∅))
1413exbidv 1837 . . . . . . . 8 (𝑦 = ∅ → (∃𝑧 𝑧:𝑦onto→∅ ↔ ∃𝑧 𝑧:∅–onto→∅))
1514rspcev 3282 . . . . . . 7 ((∅ ∈ 𝒫 𝑌 ∧ ∃𝑧 𝑧:∅–onto→∅) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto→∅)
166, 12, 15mp2an 704 . . . . . 6 𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto→∅
17 foeq3 6026 . . . . . . . 8 (𝑋 = ∅ → (𝑧:𝑦onto𝑋𝑧:𝑦onto→∅))
1817exbidv 1837 . . . . . . 7 (𝑋 = ∅ → (∃𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑧 𝑧:𝑦onto→∅))
1918rexbidv 3034 . . . . . 6 (𝑋 = ∅ → (∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto→∅))
2016, 19mpbiri 247 . . . . 5 (𝑋 = ∅ → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
2120adantl 481 . . . 4 ((𝑌 ∈ V ∧ 𝑋 = ∅) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
225, 212thd 254 . . 3 ((𝑌 ∈ V ∧ 𝑋 = ∅) → (𝑋* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
23 brwdomn0 8357 . . . . 5 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑥 𝑥:𝑌onto𝑋))
2423adantl 481 . . . 4 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑋* 𝑌 ↔ ∃𝑥 𝑥:𝑌onto𝑋))
25 foeq1 6024 . . . . . . 7 (𝑥 = 𝑧 → (𝑥:𝑌onto𝑋𝑧:𝑌onto𝑋))
2625cbvexv 2263 . . . . . 6 (∃𝑥 𝑥:𝑌onto𝑋 ↔ ∃𝑧 𝑧:𝑌onto𝑋)
27 pwidg 4121 . . . . . . . . 9 (𝑌 ∈ V → 𝑌 ∈ 𝒫 𝑌)
2827ad2antrr 758 . . . . . . . 8 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑌 ∈ 𝒫 𝑌)
29 foeq2 6025 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑧:𝑦onto𝑋𝑧:𝑌onto𝑋))
3029exbidv 1837 . . . . . . . . 9 (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
3130rspcev 3282 . . . . . . . 8 ((𝑌 ∈ 𝒫 𝑌 ∧ ∃𝑧 𝑧:𝑌onto𝑋) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
3228, 31sylancom 698 . . . . . . 7 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ ∃𝑧 𝑧:𝑌onto𝑋) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
3332ex 449 . . . . . 6 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑧 𝑧:𝑌onto𝑋 → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
3426, 33syl5bi 231 . . . . 5 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑥 𝑥:𝑌onto𝑋 → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
35 n0 3890 . . . . . . . . . . 11 (𝑋 ≠ ∅ ↔ ∃𝑤 𝑤𝑋)
3635biimpi 205 . . . . . . . . . 10 (𝑋 ≠ ∅ → ∃𝑤 𝑤𝑋)
3736ad2antlr 759 . . . . . . . . 9 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) → ∃𝑤 𝑤𝑋)
38 vex 3176 . . . . . . . . . . . . 13 𝑧 ∈ V
39 difexg 4735 . . . . . . . . . . . . . 14 (𝑌 ∈ V → (𝑌𝑦) ∈ V)
40 snex 4835 . . . . . . . . . . . . . 14 {𝑤} ∈ V
41 xpexg 6858 . . . . . . . . . . . . . 14 (((𝑌𝑦) ∈ V ∧ {𝑤} ∈ V) → ((𝑌𝑦) × {𝑤}) ∈ V)
4239, 40, 41sylancl 693 . . . . . . . . . . . . 13 (𝑌 ∈ V → ((𝑌𝑦) × {𝑤}) ∈ V)
43 unexg 6857 . . . . . . . . . . . . 13 ((𝑧 ∈ V ∧ ((𝑌𝑦) × {𝑤}) ∈ V) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
4438, 42, 43sylancr 694 . . . . . . . . . . . 12 (𝑌 ∈ V → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
4544adantr 480 . . . . . . . . . . 11 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
4645ad2antrr 758 . . . . . . . . . 10 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
47 fofn 6030 . . . . . . . . . . . . . . 15 (𝑧:𝑦onto𝑋𝑧 Fn 𝑦)
4847adantl 481 . . . . . . . . . . . . . 14 ((𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋) → 𝑧 Fn 𝑦)
4948ad2antlr 759 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → 𝑧 Fn 𝑦)
50 vex 3176 . . . . . . . . . . . . . 14 𝑤 ∈ V
51 fnconstg 6006 . . . . . . . . . . . . . 14 (𝑤 ∈ V → ((𝑌𝑦) × {𝑤}) Fn (𝑌𝑦))
5250, 51mp1i 13 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ((𝑌𝑦) × {𝑤}) Fn (𝑌𝑦))
53 disjdif 3992 . . . . . . . . . . . . . 14 (𝑦 ∩ (𝑌𝑦)) = ∅
5453a1i 11 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑦 ∩ (𝑌𝑦)) = ∅)
55 fnun 5911 . . . . . . . . . . . . 13 (((𝑧 Fn 𝑦 ∧ ((𝑌𝑦) × {𝑤}) Fn (𝑌𝑦)) ∧ (𝑦 ∩ (𝑌𝑦)) = ∅) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌𝑦)))
5649, 52, 54, 55syl21anc 1317 . . . . . . . . . . . 12 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌𝑦)))
57 elpwi 4117 . . . . . . . . . . . . . . . 16 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
58 undif 4001 . . . . . . . . . . . . . . . 16 (𝑦𝑌 ↔ (𝑦 ∪ (𝑌𝑦)) = 𝑌)
5957, 58sylib 207 . . . . . . . . . . . . . . 15 (𝑦 ∈ 𝒫 𝑌 → (𝑦 ∪ (𝑌𝑦)) = 𝑌)
6059ad2antrl 760 . . . . . . . . . . . . . 14 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) → (𝑦 ∪ (𝑌𝑦)) = 𝑌)
6160adantr 480 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑦 ∪ (𝑌𝑦)) = 𝑌)
6261fneq2d 5896 . . . . . . . . . . . 12 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ((𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌𝑦)) ↔ (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn 𝑌))
6356, 62mpbid 221 . . . . . . . . . . 11 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn 𝑌)
64 rnun 5460 . . . . . . . . . . . 12 ran (𝑧 ∪ ((𝑌𝑦) × {𝑤})) = (ran 𝑧 ∪ ran ((𝑌𝑦) × {𝑤}))
65 forn 6031 . . . . . . . . . . . . . . . 16 (𝑧:𝑦onto𝑋 → ran 𝑧 = 𝑋)
6665ad2antll 761 . . . . . . . . . . . . . . 15 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) → ran 𝑧 = 𝑋)
6766adantr 480 . . . . . . . . . . . . . 14 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ran 𝑧 = 𝑋)
6867uneq1d 3728 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (ran 𝑧 ∪ ran ((𝑌𝑦) × {𝑤})) = (𝑋 ∪ ran ((𝑌𝑦) × {𝑤})))
69 fconst6g 6007 . . . . . . . . . . . . . . . 16 (𝑤𝑋 → ((𝑌𝑦) × {𝑤}):(𝑌𝑦)⟶𝑋)
70 frn 5966 . . . . . . . . . . . . . . . 16 (((𝑌𝑦) × {𝑤}):(𝑌𝑦)⟶𝑋 → ran ((𝑌𝑦) × {𝑤}) ⊆ 𝑋)
7169, 70syl 17 . . . . . . . . . . . . . . 15 (𝑤𝑋 → ran ((𝑌𝑦) × {𝑤}) ⊆ 𝑋)
7271adantl 481 . . . . . . . . . . . . . 14 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ran ((𝑌𝑦) × {𝑤}) ⊆ 𝑋)
73 ssequn2 3748 . . . . . . . . . . . . . 14 (ran ((𝑌𝑦) × {𝑤}) ⊆ 𝑋 ↔ (𝑋 ∪ ran ((𝑌𝑦) × {𝑤})) = 𝑋)
7472, 73sylib 207 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑋 ∪ ran ((𝑌𝑦) × {𝑤})) = 𝑋)
7568, 74eqtrd 2644 . . . . . . . . . . . 12 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (ran 𝑧 ∪ ran ((𝑌𝑦) × {𝑤})) = 𝑋)
7664, 75syl5eq 2656 . . . . . . . . . . 11 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ran (𝑧 ∪ ((𝑌𝑦) × {𝑤})) = 𝑋)
77 df-fo 5810 . . . . . . . . . . 11 ((𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋 ↔ ((𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn 𝑌 ∧ ran (𝑧 ∪ ((𝑌𝑦) × {𝑤})) = 𝑋))
7863, 76, 77sylanbrc 695 . . . . . . . . . 10 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋)
79 foeq1 6024 . . . . . . . . . . 11 (𝑥 = (𝑧 ∪ ((𝑌𝑦) × {𝑤})) → (𝑥:𝑌onto𝑋 ↔ (𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋))
8079spcegv 3267 . . . . . . . . . 10 ((𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V → ((𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋 → ∃𝑥 𝑥:𝑌onto𝑋))
8146, 78, 80sylc 63 . . . . . . . . 9 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ∃𝑥 𝑥:𝑌onto𝑋)
8237, 81exlimddv 1850 . . . . . . . 8 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) → ∃𝑥 𝑥:𝑌onto𝑋)
8382expr 641 . . . . . . 7 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝑧:𝑦onto𝑋 → ∃𝑥 𝑥:𝑌onto𝑋))
8483exlimdv 1848 . . . . . 6 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ 𝑦 ∈ 𝒫 𝑌) → (∃𝑧 𝑧:𝑦onto𝑋 → ∃𝑥 𝑥:𝑌onto𝑋))
8584rexlimdva 3013 . . . . 5 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋 → ∃𝑥 𝑥:𝑌onto𝑋))
8634, 85impbid 201 . . . 4 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑥 𝑥:𝑌onto𝑋 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
8724, 86bitrd 267 . . 3 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑋* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
8822, 87pm2.61dane 2869 . 2 (𝑌 ∈ V → (𝑋* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
891, 88syl 17 1 (𝑌𝑉 → (𝑋* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583   × cxp 5036  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803   ≼* cwdom 8345 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-wdom 8347 This theorem is referenced by:  brwdom3  8370
 Copyright terms: Public domain W3C validator