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Theorem brwdom2 8519
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 3243 . 2 (𝑌𝑉𝑌 ∈ V)
2 0wdom 8516 . . . . . 6 (𝑌 ∈ V → ∅ ≼* 𝑌)
3 breq1 4688 . . . . . 6 (𝑋 = ∅ → (𝑋* 𝑌 ↔ ∅ ≼* 𝑌))
42, 3syl5ibrcom 237 . . . . 5 (𝑌 ∈ V → (𝑋 = ∅ → 𝑋* 𝑌))
54imp 444 . . . 4 ((𝑌 ∈ V ∧ 𝑋 = ∅) → 𝑋* 𝑌)
6 0elpw 4864 . . . . . . 7 ∅ ∈ 𝒫 𝑌
7 f1o0 6211 . . . . . . . 8 ∅:∅–1-1-onto→∅
8 f1ofo 6182 . . . . . . . 8 (∅:∅–1-1-onto→∅ → ∅:∅–onto→∅)
9 0ex 4823 . . . . . . . . 9 ∅ ∈ V
10 foeq1 6149 . . . . . . . . 9 (𝑧 = ∅ → (𝑧:∅–onto→∅ ↔ ∅:∅–onto→∅))
119, 10spcev 3331 . . . . . . . 8 (∅:∅–onto→∅ → ∃𝑧 𝑧:∅–onto→∅)
127, 8, 11mp2b 10 . . . . . . 7 𝑧 𝑧:∅–onto→∅
13 foeq2 6150 . . . . . . . . 9 (𝑦 = ∅ → (𝑧:𝑦onto→∅ ↔ 𝑧:∅–onto→∅))
1413exbidv 1890 . . . . . . . 8 (𝑦 = ∅ → (∃𝑧 𝑧:𝑦onto→∅ ↔ ∃𝑧 𝑧:∅–onto→∅))
1514rspcev 3340 . . . . . . 7 ((∅ ∈ 𝒫 𝑌 ∧ ∃𝑧 𝑧:∅–onto→∅) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto→∅)
166, 12, 15mp2an 708 . . . . . 6 𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto→∅
17 foeq3 6151 . . . . . . . 8 (𝑋 = ∅ → (𝑧:𝑦onto𝑋𝑧:𝑦onto→∅))
1817exbidv 1890 . . . . . . 7 (𝑋 = ∅ → (∃𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑧 𝑧:𝑦onto→∅))
1918rexbidv 3081 . . . . . 6 (𝑋 = ∅ → (∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto→∅))
2016, 19mpbiri 248 . . . . 5 (𝑋 = ∅ → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
2120adantl 481 . . . 4 ((𝑌 ∈ V ∧ 𝑋 = ∅) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
225, 212thd 255 . . 3 ((𝑌 ∈ V ∧ 𝑋 = ∅) → (𝑋* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
23 brwdomn0 8515 . . . . 5 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑥 𝑥:𝑌onto𝑋))
2423adantl 481 . . . 4 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑋* 𝑌 ↔ ∃𝑥 𝑥:𝑌onto𝑋))
25 foeq1 6149 . . . . . . 7 (𝑥 = 𝑧 → (𝑥:𝑌onto𝑋𝑧:𝑌onto𝑋))
2625cbvexv 2311 . . . . . 6 (∃𝑥 𝑥:𝑌onto𝑋 ↔ ∃𝑧 𝑧:𝑌onto𝑋)
27 pwidg 4206 . . . . . . . . 9 (𝑌 ∈ V → 𝑌 ∈ 𝒫 𝑌)
2827ad2antrr 762 . . . . . . . 8 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑌 ∈ 𝒫 𝑌)
29 foeq2 6150 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑧:𝑦onto𝑋𝑧:𝑌onto𝑋))
3029exbidv 1890 . . . . . . . . 9 (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
3130rspcev 3340 . . . . . . . 8 ((𝑌 ∈ 𝒫 𝑌 ∧ ∃𝑧 𝑧:𝑌onto𝑋) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
3228, 31sylancom 702 . . . . . . 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 232 . . . . 5 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑥 𝑥:𝑌onto𝑋 → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
35 n0 3964 . . . . . . . . . . 11 (𝑋 ≠ ∅ ↔ ∃𝑤 𝑤𝑋)
3635biimpi 206 . . . . . . . . . 10 (𝑋 ≠ ∅ → ∃𝑤 𝑤𝑋)
3736ad2antlr 763 . . . . . . . . 9 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) → ∃𝑤 𝑤𝑋)
38 vex 3234 . . . . . . . . . . . . 13 𝑧 ∈ V
39 difexg 4841 . . . . . . . . . . . . . 14 (𝑌 ∈ V → (𝑌𝑦) ∈ V)
40 snex 4938 . . . . . . . . . . . . . 14 {𝑤} ∈ V
41 xpexg 7002 . . . . . . . . . . . . . 14 (((𝑌𝑦) ∈ V ∧ {𝑤} ∈ V) → ((𝑌𝑦) × {𝑤}) ∈ V)
4239, 40, 41sylancl 695 . . . . . . . . . . . . 13 (𝑌 ∈ V → ((𝑌𝑦) × {𝑤}) ∈ V)
43 unexg 7001 . . . . . . . . . . . . 13 ((𝑧 ∈ V ∧ ((𝑌𝑦) × {𝑤}) ∈ V) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
4438, 42, 43sylancr 696 . . . . . . . . . . . 12 (𝑌 ∈ V → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
4544adantr 480 . . . . . . . . . . 11 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
4645ad2antrr 762 . . . . . . . . . 10 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
47 fofn 6155 . . . . . . . . . . . . . . 15 (𝑧:𝑦onto𝑋𝑧 Fn 𝑦)
4847adantl 481 . . . . . . . . . . . . . 14 ((𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋) → 𝑧 Fn 𝑦)
4948ad2antlr 763 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → 𝑧 Fn 𝑦)
50 vex 3234 . . . . . . . . . . . . . 14 𝑤 ∈ V
51 fnconstg 6131 . . . . . . . . . . . . . 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 4073 . . . . . . . . . . . . . 14 (𝑦 ∩ (𝑌𝑦)) = ∅
5453a1i 11 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑦 ∩ (𝑌𝑦)) = ∅)
55 fnun 6035 . . . . . . . . . . . . 13 (((𝑧 Fn 𝑦 ∧ ((𝑌𝑦) × {𝑤}) Fn (𝑌𝑦)) ∧ (𝑦 ∩ (𝑌𝑦)) = ∅) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌𝑦)))
5649, 52, 54, 55syl21anc 1365 . . . . . . . . . . . 12 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌𝑦)))
57 elpwi 4201 . . . . . . . . . . . . . . . 16 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
58 undif 4082 . . . . . . . . . . . . . . . 16 (𝑦𝑌 ↔ (𝑦 ∪ (𝑌𝑦)) = 𝑌)
5957, 58sylib 208 . . . . . . . . . . . . . . 15 (𝑦 ∈ 𝒫 𝑌 → (𝑦 ∪ (𝑌𝑦)) = 𝑌)
6059ad2antrl 764 . . . . . . . . . . . . . 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 6020 . . . . . . . . . . . 12 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ((𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌𝑦)) ↔ (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn 𝑌))
6356, 62mpbid 222 . . . . . . . . . . 11 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn 𝑌)
64 rnun 5576 . . . . . . . . . . . 12 ran (𝑧 ∪ ((𝑌𝑦) × {𝑤})) = (ran 𝑧 ∪ ran ((𝑌𝑦) × {𝑤}))
65 forn 6156 . . . . . . . . . . . . . . . 16 (𝑧:𝑦onto𝑋 → ran 𝑧 = 𝑋)
6665ad2antll 765 . . . . . . . . . . . . . . 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 3799 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (ran 𝑧 ∪ ran ((𝑌𝑦) × {𝑤})) = (𝑋 ∪ ran ((𝑌𝑦) × {𝑤})))
69 fconst6g 6132 . . . . . . . . . . . . . . . 16 (𝑤𝑋 → ((𝑌𝑦) × {𝑤}):(𝑌𝑦)⟶𝑋)
70 frn 6091 . . . . . . . . . . . . . . . 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 3819 . . . . . . . . . . . . . 14 (ran ((𝑌𝑦) × {𝑤}) ⊆ 𝑋 ↔ (𝑋 ∪ ran ((𝑌𝑦) × {𝑤})) = 𝑋)
7472, 73sylib 208 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑋 ∪ ran ((𝑌𝑦) × {𝑤})) = 𝑋)
7568, 74eqtrd 2685 . . . . . . . . . . . 12 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (ran 𝑧 ∪ ran ((𝑌𝑦) × {𝑤})) = 𝑋)
7664, 75syl5eq 2697 . . . . . . . . . . 11 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ran (𝑧 ∪ ((𝑌𝑦) × {𝑤})) = 𝑋)
77 df-fo 5932 . . . . . . . . . . 11 ((𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋 ↔ ((𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn 𝑌 ∧ ran (𝑧 ∪ ((𝑌𝑦) × {𝑤})) = 𝑋))
7863, 76, 77sylanbrc 699 . . . . . . . . . 10 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋)
79 foeq1 6149 . . . . . . . . . . 11 (𝑥 = (𝑧 ∪ ((𝑌𝑦) × {𝑤})) → (𝑥:𝑌onto𝑋 ↔ (𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋))
8079spcegv 3325 . . . . . . . . . 10 ((𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V → ((𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋 → ∃𝑥 𝑥:𝑌onto𝑋))
8146, 78, 80sylc 65 . . . . . . . . 9 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ∃𝑥 𝑥:𝑌onto𝑋)
8237, 81exlimddv 1903 . . . . . . . 8 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) → ∃𝑥 𝑥:𝑌onto𝑋)
8382expr 642 . . . . . . 7 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝑧:𝑦onto𝑋 → ∃𝑥 𝑥:𝑌onto𝑋))
8483exlimdv 1901 . . . . . 6 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ 𝑦 ∈ 𝒫 𝑌) → (∃𝑧 𝑧:𝑦onto𝑋 → ∃𝑥 𝑥:𝑌onto𝑋))
8584rexlimdva 3060 . . . . 5 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋 → ∃𝑥 𝑥:𝑌onto𝑋))
8634, 85impbid 202 . . . 4 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑥 𝑥:𝑌onto𝑋 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
8724, 86bitrd 268 . . 3 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑋* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
8822, 87pm2.61dane 2910 . 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 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  wrex 2942  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   class class class wbr 4685   × cxp 5141  ran crn 5144   Fn wfn 5921  wf 5922  ontowfo 5924  1-1-ontowf1o 5925  * cwdom 8503
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-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-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-wdom 8505
This theorem is referenced by:  brwdom3  8528
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