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Theorem wdompwdom 9042

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 9030	. . . . . 6 ⊢ Rel ≼^*
2	1	brrelex2i 5609	. . . . 5 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3	2	pwexd 5280	. . . 4 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 𝑌 ∈ V)
4		0ss 4350	. . . . 5 ⊢ ∅ ⊆ 𝑌
5	4	sspwi 4553	. . . 4 ⊢ 𝒫 ∅ ⊆ 𝒫 𝑌
6		ssdomg 8555	. . . 4 ⊢ (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌))
7	3, 5, 6	mpisyl 21	. . 3 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)
8		pweq 4555	. . . 4 ⊢ (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅)
9	8	breq1d 5076	. . 3 ⊢ (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌))
10	7, 9	syl5ibr 248	. 2 ⊢ (𝑋 = ∅ → (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
11		brwdomn0 9033	. . 3 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
12		vex 3497	. . . . 5 ⊢ 𝑧 ∈ V
13		fopwdom 8625	. . . . 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 255	. 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
17	10, 16	pm2.61ine 3100	1 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 class class class wbr 5066 –onto→wfo 6353 ≼ cdom 8507 ≼^* cwdom 9021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-dom 8511 df-wdom 9023

This theorem is referenced by: isfin32i 9787 hsmexlem1 9848 hsmexlem3 9850 gchhar 10101

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