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Theorem domssex 9125

Description: Weakening of domssex2 9124 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 8955	. 2 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
2		reldom 8948	. . 3 ⊢ Rel ≼
3	2	brrelex2i 5718	. 2 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
4		vex 3457	. . . . . . . 8 ⊢ 𝑓 ∈ V
5		f1stres 8009	. . . . . . . . 9 ⊢ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})⟶(𝐵 ∖ ran 𝑓)
6		difexg 5299	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
7	6	adantl 486	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → (𝐵 ∖ ran 𝑓) ∈ V)
8		snex 5410	. . . . . . . . . 10 ⊢ {𝒫 ∪ ran 𝐴} ∈ V
9		xpexg 7748	. . . . . . . . . 10 ⊢ (((𝐵 ∖ ran 𝑓) ∈ V ∧ {𝒫 ∪ ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}) ∈ V)
10	7, 8, 9	sylancl 597	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}) ∈ V)
11		fex2 7932	. . . . . . . . 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 1493	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})) ∈ V)
13		unexg 7741	. . . . . . . 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 598	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → (𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V)
15		cnvexg 7920	. . . . . . 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 7898	. . . . . 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 487	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝑓:𝐴–1-1→𝐵)
20		f1dm 6780	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1→𝐵 → dom 𝑓 = 𝐴)
21	4	dmex 7905	. . . . . . . . . 10 ⊢ dom 𝑓 ∈ V
22	20, 21	eqeltrrdi 2870	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
23	22	adantr 485	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
24		simpr 489	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
25		eqid 2761	. . . . . . . . 9 ⊢ ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) = ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})))
26	25	domss2 9123	. . . . . . . 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 1396	. . . . . . 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 1159	. . . . . 6 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐴 ⊆ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))))
29	27	simp1d 1158	. . . . . . 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 8962	. . . . . . 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 595	. . . . . 6 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐵 ≈ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))))
32	28, 31	jca 520	. . . . 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 3962	. . . . . 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 5112	. . . . . 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 643	. . . . 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 3556	. . . 4 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥))
37	36	ex 417	. . 3 ⊢ (𝑓:𝐴–1-1→𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥)))
38	37	exlimiv 1958	. 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

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1568 ∃wex 1807 ∈ wcel 2141 Vcvv 3453 ∖ cdif 3901 ∪ cun 3902 ⊆ wss 3904 𝒫 cpw 4561 {csn 4588 ∪ cuni 4871 class class class wbr 5108 I cid 5555 × cxp 5659 ^◡ccnv 5660 dom cdm 5661 ran crn 5662 ↾ cres 5663 ∘ ccom 5665 ⟶wf 6532 –1-1→wf1 6533 –1-1-onto→wf1o 6535 1^st c1st 7983 ≈ cen 8939 ≼ cdom 8940

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-1st 7985 df-2nd 7986 df-en 8943 df-dom 8944

This theorem is referenced by: (None)

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