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Theorem brwdom2 8430
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 3201 . 2 (𝑌𝑉𝑌 ∈ V)
2 0wdom 8427 . . . . . 6 (𝑌 ∈ V → ∅ ≼* 𝑌)
3 breq1 4621 . . . . . 6 (𝑋 = ∅ → (𝑋* 𝑌 ↔ ∅ ≼* 𝑌))
42, 3syl5ibrcom 237 . . . . 5 (𝑌 ∈ V → (𝑋 = ∅ → 𝑋* 𝑌))
54imp 445 . . . 4 ((𝑌 ∈ V ∧ 𝑋 = ∅) → 𝑋* 𝑌)
6 0elpw 4799 . . . . . . 7 ∅ ∈ 𝒫 𝑌
7 f1o0 6135 . . . . . . . 8 ∅:∅–1-1-onto→∅
8 f1ofo 6106 . . . . . . . 8 (∅:∅–1-1-onto→∅ → ∅:∅–onto→∅)
9 0ex 4755 . . . . . . . . 9 ∅ ∈ V
10 foeq1 6073 . . . . . . . . 9 (𝑧 = ∅ → (𝑧:∅–onto→∅ ↔ ∅:∅–onto→∅))
119, 10spcev 3289 . . . . . . . 8 (∅:∅–onto→∅ → ∃𝑧 𝑧:∅–onto→∅)
127, 8, 11mp2b 10 . . . . . . 7 𝑧 𝑧:∅–onto→∅
13 foeq2 6074 . . . . . . . . 9 (𝑦 = ∅ → (𝑧:𝑦onto→∅ ↔ 𝑧:∅–onto→∅))
1413exbidv 1847 . . . . . . . 8 (𝑦 = ∅ → (∃𝑧 𝑧:𝑦onto→∅ ↔ ∃𝑧 𝑧:∅–onto→∅))
1514rspcev 3298 . . . . . . 7 ((∅ ∈ 𝒫 𝑌 ∧ ∃𝑧 𝑧:∅–onto→∅) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto→∅)
166, 12, 15mp2an 707 . . . . . 6 𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto→∅
17 foeq3 6075 . . . . . . . 8 (𝑋 = ∅ → (𝑧:𝑦onto𝑋𝑧:𝑦onto→∅))
1817exbidv 1847 . . . . . . 7 (𝑋 = ∅ → (∃𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑧 𝑧:𝑦onto→∅))
1918rexbidv 3046 . . . . . 6 (𝑋 = ∅ → (∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto→∅))
2016, 19mpbiri 248 . . . . 5 (𝑋 = ∅ → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
2120adantl 482 . . . 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 8426 . . . . 5 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑥 𝑥:𝑌onto𝑋))
2423adantl 482 . . . 4 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑋* 𝑌 ↔ ∃𝑥 𝑥:𝑌onto𝑋))
25 foeq1 6073 . . . . . . 7 (𝑥 = 𝑧 → (𝑥:𝑌onto𝑋𝑧:𝑌onto𝑋))
2625cbvexv 2274 . . . . . 6 (∃𝑥 𝑥:𝑌onto𝑋 ↔ ∃𝑧 𝑧:𝑌onto𝑋)
27 pwidg 4149 . . . . . . . . 9 (𝑌 ∈ V → 𝑌 ∈ 𝒫 𝑌)
2827ad2antrr 761 . . . . . . . 8 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑌 ∈ 𝒫 𝑌)
29 foeq2 6074 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑧:𝑦onto𝑋𝑧:𝑌onto𝑋))
3029exbidv 1847 . . . . . . . . 9 (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
3130rspcev 3298 . . . . . . . 8 ((𝑌 ∈ 𝒫 𝑌 ∧ ∃𝑧 𝑧:𝑌onto𝑋) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
3228, 31sylancom 700 . . . . . . 7 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ ∃𝑧 𝑧:𝑌onto𝑋) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
3332ex 450 . . . . . 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 3912 . . . . . . . . . . 11 (𝑋 ≠ ∅ ↔ ∃𝑤 𝑤𝑋)
3635biimpi 206 . . . . . . . . . 10 (𝑋 ≠ ∅ → ∃𝑤 𝑤𝑋)
3736ad2antlr 762 . . . . . . . . 9 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) → ∃𝑤 𝑤𝑋)
38 vex 3192 . . . . . . . . . . . . 13 𝑧 ∈ V
39 difexg 4773 . . . . . . . . . . . . . 14 (𝑌 ∈ V → (𝑌𝑦) ∈ V)
40 snex 4874 . . . . . . . . . . . . . 14 {𝑤} ∈ V
41 xpexg 6920 . . . . . . . . . . . . . 14 (((𝑌𝑦) ∈ V ∧ {𝑤} ∈ V) → ((𝑌𝑦) × {𝑤}) ∈ V)
4239, 40, 41sylancl 693 . . . . . . . . . . . . 13 (𝑌 ∈ V → ((𝑌𝑦) × {𝑤}) ∈ V)
43 unexg 6919 . . . . . . . . . . . . 13 ((𝑧 ∈ V ∧ ((𝑌𝑦) × {𝑤}) ∈ V) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
4438, 42, 43sylancr 694 . . . . . . . . . . . 12 (𝑌 ∈ V → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
4544adantr 481 . . . . . . . . . . 11 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
4645ad2antrr 761 . . . . . . . . . 10 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
47 fofn 6079 . . . . . . . . . . . . . . 15 (𝑧:𝑦onto𝑋𝑧 Fn 𝑦)
4847adantl 482 . . . . . . . . . . . . . 14 ((𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋) → 𝑧 Fn 𝑦)
4948ad2antlr 762 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → 𝑧 Fn 𝑦)
50 vex 3192 . . . . . . . . . . . . . 14 𝑤 ∈ V
51 fnconstg 6055 . . . . . . . . . . . . . 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 4017 . . . . . . . . . . . . . 14 (𝑦 ∩ (𝑌𝑦)) = ∅
5453a1i 11 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑦 ∩ (𝑌𝑦)) = ∅)
55 fnun 5960 . . . . . . . . . . . . 13 (((𝑧 Fn 𝑦 ∧ ((𝑌𝑦) × {𝑤}) Fn (𝑌𝑦)) ∧ (𝑦 ∩ (𝑌𝑦)) = ∅) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌𝑦)))
5649, 52, 54, 55syl21anc 1322 . . . . . . . . . . . 12 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌𝑦)))
57 elpwi 4145 . . . . . . . . . . . . . . . 16 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
58 undif 4026 . . . . . . . . . . . . . . . 16 (𝑦𝑌 ↔ (𝑦 ∪ (𝑌𝑦)) = 𝑌)
5957, 58sylib 208 . . . . . . . . . . . . . . 15 (𝑦 ∈ 𝒫 𝑌 → (𝑦 ∪ (𝑌𝑦)) = 𝑌)
6059ad2antrl 763 . . . . . . . . . . . . . 14 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) → (𝑦 ∪ (𝑌𝑦)) = 𝑌)
6160adantr 481 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑦 ∪ (𝑌𝑦)) = 𝑌)
6261fneq2d 5945 . . . . . . . . . . . 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 5505 . . . . . . . . . . . 12 ran (𝑧 ∪ ((𝑌𝑦) × {𝑤})) = (ran 𝑧 ∪ ran ((𝑌𝑦) × {𝑤}))
65 forn 6080 . . . . . . . . . . . . . . . 16 (𝑧:𝑦onto𝑋 → ran 𝑧 = 𝑋)
6665ad2antll 764 . . . . . . . . . . . . . . 15 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) → ran 𝑧 = 𝑋)
6766adantr 481 . . . . . . . . . . . . . 14 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ran 𝑧 = 𝑋)
6867uneq1d 3749 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (ran 𝑧 ∪ ran ((𝑌𝑦) × {𝑤})) = (𝑋 ∪ ran ((𝑌𝑦) × {𝑤})))
69 fconst6g 6056 . . . . . . . . . . . . . . . 16 (𝑤𝑋 → ((𝑌𝑦) × {𝑤}):(𝑌𝑦)⟶𝑋)
70 frn 6015 . . . . . . . . . . . . . . . 16 (((𝑌𝑦) × {𝑤}):(𝑌𝑦)⟶𝑋 → ran ((𝑌𝑦) × {𝑤}) ⊆ 𝑋)
7169, 70syl 17 . . . . . . . . . . . . . . 15 (𝑤𝑋 → ran ((𝑌𝑦) × {𝑤}) ⊆ 𝑋)
7271adantl 482 . . . . . . . . . . . . . 14 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ran ((𝑌𝑦) × {𝑤}) ⊆ 𝑋)
73 ssequn2 3769 . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . 12 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (ran 𝑧 ∪ ran ((𝑌𝑦) × {𝑤})) = 𝑋)
7664, 75syl5eq 2667 . . . . . . . . . . 11 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ran (𝑧 ∪ ((𝑌𝑦) × {𝑤})) = 𝑋)
77 df-fo 5858 . . . . . . . . . . 11 ((𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋 ↔ ((𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn 𝑌 ∧ ran (𝑧 ∪ ((𝑌𝑦) × {𝑤})) = 𝑋))
7863, 76, 77sylanbrc 697 . . . . . . . . . 10 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋)
79 foeq1 6073 . . . . . . . . . . 11 (𝑥 = (𝑧 ∪ ((𝑌𝑦) × {𝑤})) → (𝑥:𝑌onto𝑋 ↔ (𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋))
8079spcegv 3283 . . . . . . . . . 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 1860 . . . . . . . 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 1858 . . . . . 6 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ 𝑦 ∈ 𝒫 𝑌) → (∃𝑧 𝑧:𝑦onto𝑋 → ∃𝑥 𝑥:𝑌onto𝑋))
8584rexlimdva 3025 . . . . 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 2877 . 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 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2908  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153   class class class wbr 4618   × cxp 5077  ran crn 5080   Fn wfn 5847  wf 5848  ontowfo 5850  1-1-ontowf1o 5851  * cwdom 8414
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 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-wdom 8416
This theorem is referenced by:  brwdom3  8439
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