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Theorem brwdomn0 9609

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 9606	. . . 4 ⊢ Rel ≼^*
2	1	brrelex2i 5742	. . 3 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3	2	a1i 11	. 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 → 𝑌 ∈ V))
4		fof 6820	. . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑧:𝑌⟶𝑋)
5	4	fdmd 6746	. . . . 5 ⊢ (𝑧:𝑌–onto→𝑋 → dom 𝑧 = 𝑌)
6		vex 3484	. . . . . 6 ⊢ 𝑧 ∈ V
7	6	dmex 7931	. . . . 5 ⊢ dom 𝑧 ∈ V
8	5, 7	eqeltrrdi 2850	. . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V)
9	8	exlimiv 1930	. . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V)
10	9	a1i 11	. 2 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V))
11		brwdom 9607	. . . 4 ⊢ (𝑌 ∈ V → (𝑋 ≼^* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
12		df-ne 2941	. . . . . 6 ⊢ (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅)
13		biorf 937	. . . . . 6 ⊢ (¬ 𝑋 = ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
14	12, 13	sylbi 217	. . . . 5 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
15	14	bicomd 223	. . . 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 379	1 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 848 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 class class class wbr 5143 dom cdm 5685 –onto→wfo 6559 ≼^* cwdom 9604

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-fn 6564 df-f 6565 df-fo 6567 df-wdom 9605

This theorem is referenced by: brwdom2 9613 wdomtr 9615 wdompwdom 9618 canthwdom 9619 wdomfil 10101 fin1a2lem7 10446

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