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Theorem brwdomn0 8886

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 8883	. . . 4 ⊢ Rel ≼^*
2	1	brrelex2i 5502	. . 3 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3	2	a1i 11	. 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 → 𝑌 ∈ V))
4		fof 6465	. . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑧:𝑌⟶𝑋)
5	4	fdmd 6398	. . . . 5 ⊢ (𝑧:𝑌–onto→𝑋 → dom 𝑧 = 𝑌)
6		vex 3443	. . . . . 6 ⊢ 𝑧 ∈ V
7	6	dmex 7479	. . . . 5 ⊢ dom 𝑧 ∈ V
8	5, 7	syl6eqelr 2894	. . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V)
9	8	exlimiv 1912	. . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V)
10	9	a1i 11	. 2 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V))
11		brwdom 8884	. . . 4 ⊢ (𝑌 ∈ V → (𝑋 ≼^* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
12		df-ne 2987	. . . . . 6 ⊢ (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅)
13		biorf 931	. . . . . 6 ⊢ (¬ 𝑋 = ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
14	12, 13	sylbi 218	. . . . 5 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
15	14	bicomd 224	. . . 4 ⊢ (𝑋 ≠ ∅ → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋) ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
16	11, 15	sylan9bbr 511	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑌 ∈ V) → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
17	16	ex 413	. 2 ⊢ (𝑋 ≠ ∅ → (𝑌 ∈ V → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)))
18	3, 10, 17	pm5.21ndd 381	1 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∨ wo 842 = wceq 1525 ∃wex 1765 ∈ wcel 2083 ≠ wne 2986 Vcvv 3440 ∅c0 4217 class class class wbr 4968 dom cdm 5450 –onto→wfo 6230 ≼^* cwdom 8874

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 ax-un 7326

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-xp 5456 df-rel 5457 df-cnv 5458 df-dm 5460 df-rn 5461 df-fn 6235 df-f 6236 df-fo 6238 df-wdom 8876

This theorem is referenced by: brwdom2 8890 wdomtr 8892 wdompwdom 8895 canthwdom 8896 wdomfil 9340 fin1a2lem7 9681

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