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Theorem brwdomn0 9258

Description: Weak dominance over nonempty sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)

**Assertion**
Ref	Expression
brwdomn0	⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))

Distinct variable groups: 𝑧,𝑋 𝑧,𝑌

**Proof of Theorem brwdomn0**
Step	Hyp	Ref	Expression
1		relwdom 9255	. . . 4 ⊢ Rel ≼^*
2	1	brrelex2i 5635	. . 3 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3	2	a1i 11	. 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 → 𝑌 ∈ V))
4		fof 6672	. . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑧:𝑌⟶𝑋)
5	4	fdmd 6595	. . . . 5 ⊢ (𝑧:𝑌–onto→𝑋 → dom 𝑧 = 𝑌)
6		vex 3426	. . . . . 6 ⊢ 𝑧 ∈ V
7	6	dmex 7732	. . . . 5 ⊢ dom 𝑧 ∈ V
8	5, 7	eqeltrrdi 2848	. . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V)
9	8	exlimiv 1934	. . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V)
10	9	a1i 11	. 2 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V))
11		brwdom 9256	. . . 4 ⊢ (𝑌 ∈ V → (𝑋 ≼^* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
12		df-ne 2943	. . . . . 6 ⊢ (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅)
13		biorf 933	. . . . . 6 ⊢ (¬ 𝑋 = ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
14	12, 13	sylbi 216	. . . . 5 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
15	14	bicomd 222	. . . 4 ⊢ (𝑋 ≠ ∅ → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋) ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
16	11, 15	sylan9bbr 510	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑌 ∈ V) → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
17	16	ex 412	. 2 ⊢ (𝑋 ≠ ∅ → (𝑌 ∈ V → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)))
18	3, 10, 17	pm5.21ndd 380	1 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 843 = wceq 1539 ∃wex 1783 ∈ wcel 2108 ≠ wne 2942 Vcvv 3422 ∅c0 4253 class class class wbr 5070 dom cdm 5580 –onto→wfo 6416 ≼^* cwdom 9253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-xp 5586 df-rel 5587 df-cnv 5588 df-dm 5590 df-rn 5591 df-fn 6421 df-f 6422 df-fo 6424 df-wdom 9254

This theorem is referenced by: brwdom2 9262 wdomtr 9264 wdompwdom 9267 canthwdom 9268 wdomfil 9748 fin1a2lem7 10093

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