brwdomn0 - Metamath Proof Explorer

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Theorem brwdomn0 9600

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⟶onto𝑋))

Distinct variable groups: 𝑧,𝑋 𝑧,𝑌

**Proof of Theorem brwdomn0**
Step	Hyp	Ref	Expression
1		relwdom 9597	. . . 4 ⊢ Rel ≼^*
2	1	brrelex2i 5715	. . 3 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3	2	a1i 11	. 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 → 𝑌 ∈ V))
4		fof 6826	. . . . . 6 ⊢ (𝑧:𝑌⟶onto𝑋 → 𝑧:𝑌⟶𝑋)
5	4	fdmd 6750	. . . . 5 ⊢ (𝑧:𝑌⟶onto𝑋 → dom 𝑧 = 𝑌)
6		vex 3458	. . . . . 6 ⊢ 𝑧 ∈ V
7	6	dmex 7955	. . . . 5 ⊢ dom 𝑧 ∈ V
8	5, 7	eqeltrrdi 2873	. . . 4 ⊢ (𝑧:𝑌⟶onto𝑋 → 𝑌 ∈ V)
9	8	exlimiv 1965	. . 3 ⊢ (∃𝑧 𝑧:𝑌⟶onto𝑋 → 𝑌 ∈ V)
10	9	a1i 11	. 2 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌⟶onto𝑋 → 𝑌 ∈ V))
11		brwdom 9598	. . . 4 ⊢ (𝑌 ∈ V → (𝑋 ≼^* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌⟶onto𝑋)))
12		df-ne 2960	. . . . . 6 ⊢ (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅)
13		biorf 950	. . . . . 6 ⊢ (¬ 𝑋 = ∅ → (∃𝑧 𝑧:𝑌⟶onto𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌⟶onto𝑋)))
14	12, 13	sylbi 220	. . . . 5 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌⟶onto𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌⟶onto𝑋)))
15	14	bicomd 226	. . . 4 ⊢ (𝑋 ≠ ∅ → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌⟶onto𝑋) ↔ ∃𝑧 𝑧:𝑌⟶onto𝑋))
16	11, 15	sylan9bbr 520	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑌 ∈ V) → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌⟶onto𝑋))
17	16	ex 418	. 2 ⊢ (𝑋 ≠ ∅ → (𝑌 ∈ V → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌⟶onto𝑋)))
18	3, 10, 17	pm5.21ndd 382	1 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌⟶onto𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 861 = wceq 1570 ∃wex 1812 ∈ wcel 2147 ≠ wne 2959 Vcvv 3454 ∅c0 4281 class class class wbr 5108 dom cdm 5658 ⟶ontowfo 6564 ≼^* cwdom 9595

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2102 df-clab 2742 df-cleq 2755 df-clel 2839 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-opab 5174 df-xp 5664 df-rel 5665 df-cnv 5666 df-dm 5668 df-rn 5669 df-fn 6569 df-f 6570 df-fo 6572 df-wdom 9596

This theorem is used by: brwdom2 9604 wdomtr 9606 wdompwdom 9609 canthwdom 9610 wdomfil 10179 fin1a2lem7 10523

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