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Theorem wdompwdom 9183

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 9171	. . . . . 6 ⊢ Rel ≼^*
2	1	brrelex2i 5595	. . . . 5 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3	2	pwexd 5261	. . . 4 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 𝑌 ∈ V)
4		0ss 4301	. . . . 5 ⊢ ∅ ⊆ 𝑌
5	4	sspwi 4517	. . . 4 ⊢ 𝒫 ∅ ⊆ 𝒫 𝑌
6		ssdomg 8663	. . . 4 ⊢ (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌))
7	3, 5, 6	mpisyl 21	. . 3 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)
8		pweq 4519	. . . 4 ⊢ (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅)
9	8	breq1d 5053	. . 3 ⊢ (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌))
10	7, 9	syl5ibr 249	. 2 ⊢ (𝑋 = ∅ → (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
11		brwdomn0 9174	. . 3 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
12		vex 3405	. . . . 5 ⊢ 𝑧 ∈ V
13		fopwdom 8742	. . . . 5 ⊢ ((𝑧 ∈ V ∧ 𝑧:𝑌–onto→𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌)
14	12, 13	mpan 690	. . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
15	14	exlimiv 1938	. . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
16	11, 15	syl6bi 256	. 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
17	10, 16	pm2.61ine 3018	1 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1543 ∃wex 1787 ∈ wcel 2110 ≠ wne 2935 Vcvv 3401 ⊆ wss 3857 ∅c0 4227 𝒫 cpw 4503 class class class wbr 5043 –onto→wfo 6367 ≼ cdom 8613 ≼^* cwdom 9169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-dom 8617 df-wdom 9170

This theorem is referenced by: isfin32i 9962 hsmexlem1 10023 hsmexlem3 10025 gchhar 10276

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