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Theorem brwdom2 8655
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 3369 . 2 (𝑌𝑉𝑌 ∈ V)
2 0wdom 8652 . . . . . 6 (𝑌 ∈ V → ∅ ≼* 𝑌)
3 breq1 4800 . . . . . 6 (𝑋 = ∅ → (𝑋* 𝑌 ↔ ∅ ≼* 𝑌))
42, 3syl5ibrcom 238 . . . . 5 (𝑌 ∈ V → (𝑋 = ∅ → 𝑋* 𝑌))
54imp 394 . . . 4 ((𝑌 ∈ V ∧ 𝑋 = ∅) → 𝑋* 𝑌)
6 0elpw 4979 . . . . . . 7 ∅ ∈ 𝒫 𝑌
7 f1o0 6329 . . . . . . . 8 ∅:∅–1-1-onto→∅
8 f1ofo 6300 . . . . . . . 8 (∅:∅–1-1-onto→∅ → ∅:∅–onto→∅)
9 0ex 4937 . . . . . . . . 9 ∅ ∈ V
10 foeq1 6267 . . . . . . . . 9 (𝑧 = ∅ → (𝑧:∅–onto→∅ ↔ ∅:∅–onto→∅))
119, 10spcev 3456 . . . . . . . 8 (∅:∅–onto→∅ → ∃𝑧 𝑧:∅–onto→∅)
127, 8, 11mp2b 10 . . . . . . 7 𝑧 𝑧:∅–onto→∅
13 foeq2 6268 . . . . . . . . 9 (𝑦 = ∅ → (𝑧:𝑦onto→∅ ↔ 𝑧:∅–onto→∅))
1413exbidv 2005 . . . . . . . 8 (𝑦 = ∅ → (∃𝑧 𝑧:𝑦onto→∅ ↔ ∃𝑧 𝑧:∅–onto→∅))
1514rspcev 3465 . . . . . . 7 ((∅ ∈ 𝒫 𝑌 ∧ ∃𝑧 𝑧:∅–onto→∅) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto→∅)
166, 12, 15mp2an 673 . . . . . 6 𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto→∅
17 foeq3 6269 . . . . . . . 8 (𝑋 = ∅ → (𝑧:𝑦onto𝑋𝑧:𝑦onto→∅))
1817exbidv 2005 . . . . . . 7 (𝑋 = ∅ → (∃𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑧 𝑧:𝑦onto→∅))
1918rexbidv 3204 . . . . . 6 (𝑋 = ∅ → (∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto→∅))
2016, 19mpbiri 249 . . . . 5 (𝑋 = ∅ → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
2120adantl 468 . . . 4 ((𝑌 ∈ V ∧ 𝑋 = ∅) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
225, 212thd 256 . . 3 ((𝑌 ∈ V ∧ 𝑋 = ∅) → (𝑋* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
23 brwdomn0 8651 . . . . 5 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑥 𝑥:𝑌onto𝑋))
2423adantl 468 . . . 4 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑋* 𝑌 ↔ ∃𝑥 𝑥:𝑌onto𝑋))
25 foeq1 6267 . . . . . . 7 (𝑥 = 𝑧 → (𝑥:𝑌onto𝑋𝑧:𝑌onto𝑋))
2625cbvexvw 2129 . . . . . 6 (∃𝑥 𝑥:𝑌onto𝑋 ↔ ∃𝑧 𝑧:𝑌onto𝑋)
27 pwidg 4322 . . . . . . . . 9 (𝑌 ∈ V → 𝑌 ∈ 𝒫 𝑌)
2827ad2antrr 706 . . . . . . . 8 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑌 ∈ 𝒫 𝑌)
29 foeq2 6268 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑧:𝑦onto𝑋𝑧:𝑌onto𝑋))
3029exbidv 2005 . . . . . . . . 9 (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
3130rspcev 3465 . . . . . . . 8 ((𝑌 ∈ 𝒫 𝑌 ∧ ∃𝑧 𝑧:𝑌onto𝑋) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
3228, 31sylancom 577 . . . . . . 7 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ ∃𝑧 𝑧:𝑌onto𝑋) → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋)
3332ex 398 . . . . . 6 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑧 𝑧:𝑌onto𝑋 → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
3426, 33syl5bi 233 . . . . 5 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑥 𝑥:𝑌onto𝑋 → ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
35 n0 4089 . . . . . . . . . . 11 (𝑋 ≠ ∅ ↔ ∃𝑤 𝑤𝑋)
3635biimpi 207 . . . . . . . . . 10 (𝑋 ≠ ∅ → ∃𝑤 𝑤𝑋)
3736ad2antlr 707 . . . . . . . . 9 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) → ∃𝑤 𝑤𝑋)
38 vex 3358 . . . . . . . . . . . . 13 𝑧 ∈ V
39 difexg 4956 . . . . . . . . . . . . . 14 (𝑌 ∈ V → (𝑌𝑦) ∈ V)
40 snex 5050 . . . . . . . . . . . . . 14 {𝑤} ∈ V
41 xpexg 7128 . . . . . . . . . . . . . 14 (((𝑌𝑦) ∈ V ∧ {𝑤} ∈ V) → ((𝑌𝑦) × {𝑤}) ∈ V)
4239, 40, 41sylancl 575 . . . . . . . . . . . . 13 (𝑌 ∈ V → ((𝑌𝑦) × {𝑤}) ∈ V)
43 unexg 7127 . . . . . . . . . . . . 13 ((𝑧 ∈ V ∧ ((𝑌𝑦) × {𝑤}) ∈ V) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
4438, 42, 43sylancr 576 . . . . . . . . . . . 12 (𝑌 ∈ V → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
4544adantr 467 . . . . . . . . . . 11 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
4645ad2antrr 706 . . . . . . . . . 10 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V)
47 fofn 6273 . . . . . . . . . . . . . . 15 (𝑧:𝑦onto𝑋𝑧 Fn 𝑦)
4847adantl 468 . . . . . . . . . . . . . 14 ((𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋) → 𝑧 Fn 𝑦)
4948ad2antlr 707 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → 𝑧 Fn 𝑦)
50 vex 3358 . . . . . . . . . . . . . 14 𝑤 ∈ V
51 fnconstg 6248 . . . . . . . . . . . . . 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 4192 . . . . . . . . . . . . . 14 (𝑦 ∩ (𝑌𝑦)) = ∅
5453a1i 11 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑦 ∩ (𝑌𝑦)) = ∅)
55 fnun 6148 . . . . . . . . . . . . 13 (((𝑧 Fn 𝑦 ∧ ((𝑌𝑦) × {𝑤}) Fn (𝑌𝑦)) ∧ (𝑦 ∩ (𝑌𝑦)) = ∅) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌𝑦)))
5649, 52, 54, 55syl21anc 855 . . . . . . . . . . . 12 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌𝑦)))
57 elpwi 4317 . . . . . . . . . . . . . . . 16 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
58 undif 4201 . . . . . . . . . . . . . . . 16 (𝑦𝑌 ↔ (𝑦 ∪ (𝑌𝑦)) = 𝑌)
5957, 58sylib 209 . . . . . . . . . . . . . . 15 (𝑦 ∈ 𝒫 𝑌 → (𝑦 ∪ (𝑌𝑦)) = 𝑌)
6059ad2antrl 708 . . . . . . . . . . . . . 14 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) → (𝑦 ∪ (𝑌𝑦)) = 𝑌)
6160adantr 467 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑦 ∪ (𝑌𝑦)) = 𝑌)
6261fneq2d 6133 . . . . . . . . . . . 12 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ((𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌𝑦)) ↔ (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn 𝑌))
6356, 62mpbid 223 . . . . . . . . . . 11 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn 𝑌)
64 rnun 5693 . . . . . . . . . . . 12 ran (𝑧 ∪ ((𝑌𝑦) × {𝑤})) = (ran 𝑧 ∪ ran ((𝑌𝑦) × {𝑤}))
65 forn 6274 . . . . . . . . . . . . . . . 16 (𝑧:𝑦onto𝑋 → ran 𝑧 = 𝑋)
6665ad2antll 709 . . . . . . . . . . . . . . 15 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) → ran 𝑧 = 𝑋)
6766adantr 467 . . . . . . . . . . . . . 14 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ran 𝑧 = 𝑋)
6867uneq1d 3924 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (ran 𝑧 ∪ ran ((𝑌𝑦) × {𝑤})) = (𝑋 ∪ ran ((𝑌𝑦) × {𝑤})))
69 fconst6g 6249 . . . . . . . . . . . . . . . 16 (𝑤𝑋 → ((𝑌𝑦) × {𝑤}):(𝑌𝑦)⟶𝑋)
7069frnd 6203 . . . . . . . . . . . . . . 15 (𝑤𝑋 → ran ((𝑌𝑦) × {𝑤}) ⊆ 𝑋)
7170adantl 468 . . . . . . . . . . . . . 14 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ran ((𝑌𝑦) × {𝑤}) ⊆ 𝑋)
72 ssequn2 3944 . . . . . . . . . . . . . 14 (ran ((𝑌𝑦) × {𝑤}) ⊆ 𝑋 ↔ (𝑋 ∪ ran ((𝑌𝑦) × {𝑤})) = 𝑋)
7371, 72sylib 209 . . . . . . . . . . . . 13 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑋 ∪ ran ((𝑌𝑦) × {𝑤})) = 𝑋)
7468, 73eqtrd 2808 . . . . . . . . . . . 12 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (ran 𝑧 ∪ ran ((𝑌𝑦) × {𝑤})) = 𝑋)
7564, 74syl5eq 2820 . . . . . . . . . . 11 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ran (𝑧 ∪ ((𝑌𝑦) × {𝑤})) = 𝑋)
76 df-fo 6048 . . . . . . . . . . 11 ((𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋 ↔ ((𝑧 ∪ ((𝑌𝑦) × {𝑤})) Fn 𝑌 ∧ ran (𝑧 ∪ ((𝑌𝑦) × {𝑤})) = 𝑋))
7763, 75, 76sylanbrc 573 . . . . . . . . . 10 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → (𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋)
78 foeq1 6267 . . . . . . . . . . 11 (𝑥 = (𝑧 ∪ ((𝑌𝑦) × {𝑤})) → (𝑥:𝑌onto𝑋 ↔ (𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋))
7978spcegv 3450 . . . . . . . . . 10 ((𝑧 ∪ ((𝑌𝑦) × {𝑤})) ∈ V → ((𝑧 ∪ ((𝑌𝑦) × {𝑤})):𝑌onto𝑋 → ∃𝑥 𝑥:𝑌onto𝑋))
8046, 77, 79sylc 65 . . . . . . . . 9 ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) ∧ 𝑤𝑋) → ∃𝑥 𝑥:𝑌onto𝑋)
8137, 80exlimddv 2018 . . . . . . . 8 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌𝑧:𝑦onto𝑋)) → ∃𝑥 𝑥:𝑌onto𝑋)
8281expr 445 . . . . . . 7 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝑧:𝑦onto𝑋 → ∃𝑥 𝑥:𝑌onto𝑋))
8382exlimdv 2016 . . . . . 6 (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ 𝑦 ∈ 𝒫 𝑌) → (∃𝑧 𝑧:𝑦onto𝑋 → ∃𝑥 𝑥:𝑌onto𝑋))
8483rexlimdva 3183 . . . . 5 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋 → ∃𝑥 𝑥:𝑌onto𝑋))
8534, 84impbid 203 . . . 4 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑥 𝑥:𝑌onto𝑋 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
8624, 85bitrd 269 . . 3 ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑋* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
8722, 86pm2.61dane 3033 . 2 (𝑌 ∈ V → (𝑋* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
881, 87syl 17 1 (𝑌𝑉 → (𝑋* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 383   = wceq 1634  wex 1855  wcel 2148  wne 2946  wrex 3065  Vcvv 3355  cdif 3726  cun 3727  cin 3728  wss 3729  c0 4073  𝒫 cpw 4307  {csn 4326   class class class wbr 4797   × cxp 5261  ran crn 5264   Fn wfn 6037  ontowfo 6040  1-1-ontowf1o 6041  * cwdom 8639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-wdom 8641
This theorem is referenced by:  brwdom3  8664
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