brwdomn0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > brwdomn0		Structured version Visualization version GIF version

Theorem brwdomn0 9560

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 9557	. . . 4 ⊢ Rel ≼^*
2	1	brrelex2i 5731	. . 3 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3	2	a1i 11	. 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 → 𝑌 ∈ V))
4		fof 6802	. . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑧:𝑌⟶𝑋)
5	4	fdmd 6725	. . . . 5 ⊢ (𝑧:𝑌–onto→𝑋 → dom 𝑧 = 𝑌)
6		vex 3479	. . . . . 6 ⊢ 𝑧 ∈ V
7	6	dmex 7897	. . . . 5 ⊢ dom 𝑧 ∈ V
8	5, 7	eqeltrrdi 2843	. . . 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 9558	. . . 4 ⊢ (𝑌 ∈ V → (𝑋 ≼^* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
12		df-ne 2942	. . . . . 6 ⊢ (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅)
13		biorf 936	. . . . . 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 512	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑌 ∈ V) → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
17	16	ex 414	. 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 205 ∨ wo 846 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∅c0 4321 class class class wbr 5147 dom cdm 5675 –onto→wfo 6538 ≼^* cwdom 9555

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 5298 ax-nul 5305 ax-pr 5426 ax-un 7720

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-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-dm 5685 df-rn 5686 df-fn 6543 df-f 6544 df-fo 6546 df-wdom 9556

This theorem is referenced by: brwdom2 9564 wdomtr 9566 wdompwdom 9569 canthwdom 9570 wdomfil 10052 fin1a2lem7 10397

W3C validator