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Theorem wdompwdom 9381

Description: Weak dominance strengthens to usual dominance on the power sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)

**Assertion**
Ref	Expression
wdompwdom	⊢ (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)

Proof of Theorem wdompwdom Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relwdom 9369	. . . . . 6 ⊢ Rel ≼^*
2	1	brrelex2i 5655	. . . . 5 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3	2	pwexd 5311	. . . 4 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 𝑌 ∈ V)
4		0ss 4336	. . . . 5 ⊢ ∅ ⊆ 𝑌
5	4	sspwi 4551	. . . 4 ⊢ 𝒫 ∅ ⊆ 𝒫 𝑌
6		ssdomg 8821	. . . 4 ⊢ (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌))
7	3, 5, 6	mpisyl 21	. . 3 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)
8		pweq 4553	. . . 4 ⊢ (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅)
9	8	breq1d 5091	. . 3 ⊢ (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌))
10	7, 9	syl5ibr 246	. 2 ⊢ (𝑋 = ∅ → (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
11		brwdomn0 9372	. . 3 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
12		vex 3441	. . . . 5 ⊢ 𝑧 ∈ V
13		fopwdom 8905	. . . . 5 ⊢ ((𝑧 ∈ V ∧ 𝑧:𝑌–onto→𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌)
14	12, 13	mpan 688	. . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
15	14	exlimiv 1931	. . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
16	11, 15	syl6bi 253	. 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
17	10, 16	pm2.61ine 3026	1 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∃wex 1779 ∈ wcel 2104 ≠ wne 2941 Vcvv 3437 ⊆ wss 3892 ∅c0 4262 𝒫 cpw 4539 class class class wbr 5081 –onto→wfo 6456 ≼ cdom 8762 ≼^* cwdom 9367

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-dom 8766 df-wdom 9368

This theorem is referenced by: isfin32i 10167 hsmexlem1 10228 hsmexlem3 10230 gchhar 10481

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