domssex - Metamath Proof Explorer

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

Theorem domssex 9135

Description: Weakening of domssex2 9134 to forget the functions in favor of dominance and equinumerosity. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)

**Assertion**
Ref	Expression
domssex	⊢ (𝐴 ≼ 𝐵 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

Proof of Theorem domssex Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 8964	. 2 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
2		reldom 8957	. . 3 ⊢ Rel ≼
3	2	brrelex2i 5704	. 2 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
4		vex 3454	. . . . . . . 8 ⊢ 𝑓 ∈ V
5		f1stres 8008	. . . . . . . . 9 ⊢ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})⟶(𝐵 ∖ ran 𝑓)
6		difexg 5290	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
7	6	adantl 487	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → (𝐵 ∖ ran 𝑓) ∈ V)
8		snex 5396	. . . . . . . . . 10 ⊢ {𝒫 ∪ ran 𝐴} ∈ V
9		xpexg 7747	. . . . . . . . . 10 ⊢ (((𝐵 ∖ ran 𝑓) ∈ V ∧ {𝒫 ∪ ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}) ∈ V)
10	7, 8, 9	sylancl 598	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}) ∈ V)
11		fex2 7931	. . . . . . . . 9 ⊢ (((1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})⟶(𝐵 ∖ ran 𝑓) ∧ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}) ∈ V ∧ (𝐵 ∖ ran 𝑓) ∈ V) → (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})) ∈ V)
12	5, 10, 7, 11	mp3an2i 1495	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})) ∈ V)
13		unexg 7743	. . . . . . . 8 ⊢ ((𝑓 ∈ V ∧ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})) ∈ V) → (𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V)
14	4, 12, 13	sylancr 599	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → (𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V)
15		cnvexg 7919	. . . . . . 7 ⊢ ((𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V → ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V)
16	14, 15	syl 18	. . . . . 6 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V)
17		rnexg 7897	. . . . . 6 ⊢ (^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V → ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V)
18	16, 17	syl 18	. . . . 5 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V)
19		simpl 488	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝑓:𝐴–1-1→𝐵)
20		f1dm 6772	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1→𝐵 → dom 𝑓 = 𝐴)
21	4	dmex 7904	. . . . . . . . . 10 ⊢ dom 𝑓 ∈ V
22	20, 21	eqeltrrdi 2869	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
23	22	adantr 486	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
24		simpr 490	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
25		eqid 2760	. . . . . . . . 9 ⊢ ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) = ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})))
26	25	domss2 9133	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∧ 𝐴 ⊆ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∧ (^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∘ 𝑓) = ( I ↾ 𝐴)))
27	19, 23, 24, 26	syl3anc 1398	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → (^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∧ 𝐴 ⊆ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∧ (^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∘ 𝑓) = ( I ↾ 𝐴)))
28	27	simp2d 1161	. . . . . 6 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐴 ⊆ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))))
29	27	simp1d 1160	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))))
30		f1oen3g 8971	. . . . . . 7 ⊢ ((^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V ∧ ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})))) → 𝐵 ≈ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))))
31	16, 29, 30	syl2anc 596	. . . . . 6 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐵 ≈ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))))
32	28, 31	jca 521	. . . . 5 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → (𝐴 ⊆ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∧ 𝐵 ≈ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})))))
33		sseq2 3956	. . . . . 6 ⊢ (𝑥 = ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})))))
34		breq2 5106	. . . . . 6 ⊢ (𝑥 = ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) → (𝐵 ≈ 𝑥 ↔ 𝐵 ≈ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})))))
35	33, 34	anbi12d 644	. . . . 5 ⊢ (𝑥 = ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) → ((𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥) ↔ (𝐴 ⊆ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∧ 𝐵 ≈ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))))))
36	18, 32, 35	spcedv 3552	. . . 4 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥))
37	36	ex 418	. . 3 ⊢ (𝑓:𝐴–1-1→𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥)))
38	37	exlimiv 1963	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥)))
39	1, 3, 38	sylc 66	1 ⊢ (𝐴 ≼ 𝐵 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2145 Vcvv 3450 ∖ cdif 3895 ∪ cun 3896 ⊆ wss 3898 𝒫 cpw 4556 {csn 4583 ∪ cuni 4866 class class class wbr 5102 I cid 5541 × cxp 5645 ^◡ccnv 5646 dom cdm 5647 ran crn 5648 ↾ cres 5649 ∘ ccom 5651 ⟶wf 6523 –1-1→wf1 6524 –1-1-onto→wf1o 6526 1^st c1st 7982 ≈ cen 8948 ≼ cdom 8949

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-1st 7984 df-2nd 7985 df-en 8952 df-dom 8953

This theorem is used by: (None)

W3C validator