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Theorem brwdom 9511
Description: Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.)
Assertion
Ref Expression
brwdom (𝑌𝑉 → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
Distinct variable groups:   𝑧,𝑋   𝑧,𝑌
Allowed substitution hint:   𝑉(𝑧)

Proof of Theorem brwdom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3465 . 2 (𝑌𝑉𝑌 ∈ V)
2 relwdom 9510 . . . . 5 Rel ≼*
32brrelex1i 5692 . . . 4 (𝑋* 𝑌𝑋 ∈ V)
43a1i 11 . . 3 (𝑌 ∈ V → (𝑋* 𝑌𝑋 ∈ V))
5 0ex 5268 . . . . . 6 ∅ ∈ V
6 eleq1a 2829 . . . . . 6 (∅ ∈ V → (𝑋 = ∅ → 𝑋 ∈ V))
75, 6ax-mp 5 . . . . 5 (𝑋 = ∅ → 𝑋 ∈ V)
8 forn 6763 . . . . . . 7 (𝑧:𝑌onto𝑋 → ran 𝑧 = 𝑋)
9 vex 3451 . . . . . . . 8 𝑧 ∈ V
109rnex 7853 . . . . . . 7 ran 𝑧 ∈ V
118, 10eqeltrrdi 2843 . . . . . 6 (𝑧:𝑌onto𝑋𝑋 ∈ V)
1211exlimiv 1934 . . . . 5 (∃𝑧 𝑧:𝑌onto𝑋𝑋 ∈ V)
137, 12jaoi 856 . . . 4 ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑋 ∈ V)
1413a1i 11 . . 3 (𝑌 ∈ V → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑋 ∈ V))
15 eqeq1 2737 . . . . . 6 (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅))
16 foeq3 6758 . . . . . . 7 (𝑥 = 𝑋 → (𝑧:𝑦onto𝑥𝑧:𝑦onto𝑋))
1716exbidv 1925 . . . . . 6 (𝑥 = 𝑋 → (∃𝑧 𝑧:𝑦onto𝑥 ↔ ∃𝑧 𝑧:𝑦onto𝑋))
1815, 17orbi12d 918 . . . . 5 (𝑥 = 𝑋 → ((𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑋)))
19 foeq2 6757 . . . . . . 7 (𝑦 = 𝑌 → (𝑧:𝑦onto𝑋𝑧:𝑌onto𝑋))
2019exbidv 1925 . . . . . 6 (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
2120orbi2d 915 . . . . 5 (𝑦 = 𝑌 → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑋) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
22 df-wdom 9509 . . . . 5 * = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
2318, 21, 22brabg 5500 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
2423expcom 415 . . 3 (𝑌 ∈ V → (𝑋 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))))
254, 14, 24pm5.21ndd 381 . 2 (𝑌 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
261, 25syl 17 1 (𝑌𝑉 → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 846   = wceq 1542  wex 1782  wcel 2107  Vcvv 3447  c0 4286   class class class wbr 5109  ran crn 5638  ontowfo 6498  * cwdom 9508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648  df-fn 6503  df-fo 6506  df-wdom 9509
This theorem is referenced by:  brwdomi  9512  brwdomn0  9513  0wdom  9514  fowdom  9515  domwdom  9518  wdomnumr  10008
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