wdompwdom - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > wdompwdom		Structured version Visualization version GIF version

Theorem wdompwdom 9028

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 9016	. . . . . 6 ⊢ Rel ≼^*
2	1	brrelex2i 5595	. . . . 5 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3	2	pwexd 5266	. . . 4 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 𝑌 ∈ V)
4		0ss 4336	. . . . 5 ⊢ ∅ ⊆ 𝑌
5	4	sspwi 4539	. . . 4 ⊢ 𝒫 ∅ ⊆ 𝒫 𝑌
6		ssdomg 8541	. . . 4 ⊢ (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌))
7	3, 5, 6	mpisyl 21	. . 3 ⊢ (𝑋 ≼^* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)
8		pweq 4541	. . . 4 ⊢ (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅)
9	8	breq1d 5062	. . 3 ⊢ (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌))
10	7, 9	syl5ibr 248	. 2 ⊢ (𝑋 = ∅ → (𝑋 ≼^* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
11		brwdomn0 9019	. . 3 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼^* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
12		vex 3489	. . . . 5 ⊢ 𝑧 ∈ V
13		fopwdom 8611	. . . . 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 3486 ⊆ wss 3924 ∅c0 4279 𝒫 cpw 4525 class class class wbr 5052 –onto→wfo 6339 ≼ cdom 8493 ≼^* cwdom 9007

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 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447

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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-dom 8497 df-wdom 9009

This theorem is referenced by: isfin32i 9773 hsmexlem1 9834 hsmexlem3 9836 gchhar 10087

W3C validator