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Theorem wdompwdom 9298

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 9286	. . . . . 6 ⊢ Rel ≼^*
2	1	brrelex2i 5643	. . . . 5 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3	2	pwexd 5305	. . . 4 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 𝑌 ∈ V)
4		0ss 4335	. . . . 5 ⊢ ∅ ⊆ 𝑌
5	4	sspwi 4552	. . . 4 ⊢ 𝒫 ∅ ⊆ 𝒫 𝑌
6		ssdomg 8757	. . . 4 ⊢ (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌))
7	3, 5, 6	mpisyl 21	. . 3 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)
8		pweq 4554	. . . 4 ⊢ (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅)
9	8	breq1d 5088	. . 3 ⊢ (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌))
10	7, 9	syl5ibr 245	. 2 ⊢ (𝑋 = ∅ → (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
11		brwdomn0 9289	. . 3 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
12		vex 3434	. . . . 5 ⊢ 𝑧 ∈ V
13		fopwdom 8836	. . . . 5 ⊢ ((𝑧 ∈ V ∧ 𝑧:𝑌–onto→𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌)
14	12, 13	mpan 686	. . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
15	14	exlimiv 1936	. . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
16	11, 15	syl6bi 252	. 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
17	10, 16	pm2.61ine 3029	1 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∃wex 1785 ∈ wcel 2109 ≠ wne 2944 Vcvv 3430 ⊆ wss 3891 ∅c0 4261 𝒫 cpw 4538 class class class wbr 5078 –onto→wfo 6428 ≼ cdom 8705 ≼^* cwdom 9284

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-dom 8709 df-wdom 9285

This theorem is referenced by: isfin32i 10105 hsmexlem1 10166 hsmexlem3 10168 gchhar 10419

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