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Theorem domssex 9185

Description: Weakening of domssex2 9184 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 9014	. 2 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴⟶1-1𝐵)
2		reldom 9007	. . 3 ⊢ Rel ≼
3	2	brrelex2i 5712	. 2 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
4		vex 3458	. . . . . . . 8 ⊢ 𝑓 ∈ V
5		f1stres 8049	. . . . . . . . 9 ⊢ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})⟶(𝐵 ∖ ran 𝑓)
6		difexg 5295	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
7	6	adantl 487	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝐵 ∈ V) → (𝐵 ∖ ran 𝑓) ∈ V)
8		snex 5401	. . . . . . . . . 10 ⊢ {𝒫 ∪ ran 𝐴} ∈ V
9		xpexg 7788	. . . . . . . . . 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⟶1-1𝐵 ∧ 𝐵 ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}) ∈ V)
11		fex2 7972	. . . . . . . . 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 1496	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝐵 ∈ V) → (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})) ∈ V)
13		unexg 7784	. . . . . . . 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⟶1-1𝐵 ∧ 𝐵 ∈ V) → (𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V)
15		cnvexg 7960	. . . . . . 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⟶1-1𝐵 ∧ 𝐵 ∈ V) → ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V)
17		rnexg 7938	. . . . . 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⟶1-1𝐵 ∧ 𝐵 ∈ V) → ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V)
19		simpl 488	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝐵 ∈ V) → 𝑓:𝐴⟶1-1𝐵)
20		f1dm 6806	. . . . . . . . . 10 ⊢ (𝑓:𝐴⟶1-1𝐵 → dom 𝑓 = 𝐴)
21	4	dmex 7945	. . . . . . . . . 10 ⊢ dom 𝑓 ∈ V
22	20, 21	eqeltrrdi 2873	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶1-1𝐵 → 𝐴 ∈ V)
23	22	adantr 486	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
24		simpr 490	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
25		eqid 2763	. . . . . . . . 9 ⊢ ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) = ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})))
26	25	domss2 9183	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))):𝐵⟶1-1ontoran ^◡(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 1399	. . . . . . 7 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝐵 ∈ V) → (^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))):𝐵⟶1-1ontoran ^◡(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⟶1-1𝐵 ∧ 𝐵 ∈ V) → 𝐴 ⊆ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))))
29	27	simp1d 1160	. . . . . . 7 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝐵 ∈ V) → ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))):𝐵⟶1-1ontoran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))))
30		f1oen3g 9021	. . . . . . 7 ⊢ ((^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V ∧ ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))):𝐵⟶1-1ontoran ^◡(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⟶1-1𝐵 ∧ 𝐵 ∈ V) → 𝐵 ≈ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))))
32	28, 31	jca 521	. . . . 5 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 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 3939	. . . . . 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 5110	. . . . . 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 3556	. . . 4 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝐵 ∈ V) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥))
37	36	ex 418	. . 3 ⊢ (𝑓:𝐴⟶1-1𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥)))
38	37	exlimiv 1965	. 2 ⊢ (∃𝑓 𝑓:𝐴⟶1-1𝐵 → (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 2147 Vcvv 3454 ⊆ wss 3900 ∖ cdif 4048 ∪ cun 4152 𝒫 cpw 4560 {csn 4588 ∪ cuni 4870 class class class wbr 5106 I cid 5549 × cxp 5653 ^◡ccnv 5654 dom cdm 5655 ran crn 5656 ↾ cres 5657 ∘ ccom 5659 ⟶wf 6554 ⟶1-1wf1 6555 ⟶1-1ontowf1o 6557 1^st c1st 8023 ≈ cen 8998 ≼ cdom 8999

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-1st 8025 df-2nd 8026 df-en 9002 df-dom 9003

This theorem is used by: (None)

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