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Theorem domssex 9082

Description: Weakening of domssex2 9081 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 8898	. 2 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
2		reldom 8889	. . 3 ⊢ Rel ≼
3	2	brrelex2i 5689	. 2 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
4		vex 3449	. . . . . . . 8 ⊢ 𝑓 ∈ V
5		f1stres 7945	. . . . . . . . 9 ⊢ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})⟶(𝐵 ∖ ran 𝑓)
6		difexg 5284	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
7	6	adantl 482	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → (𝐵 ∖ ran 𝑓) ∈ V)
8		snex 5388	. . . . . . . . . 10 ⊢ {𝒫 ∪ ran 𝐴} ∈ V
9		xpexg 7684	. . . . . . . . . 10 ⊢ (((𝐵 ∖ ran 𝑓) ∈ V ∧ {𝒫 ∪ ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}) ∈ V)
10	7, 8, 9	sylancl 586	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}) ∈ V)
11		fex2 7870	. . . . . . . . 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 1466	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})) ∈ V)
13		unexg 7683	. . . . . . . 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 587	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → (𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V)
15		cnvexg 7861	. . . . . . 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 17	. . . . . 6 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V)
17		rnexg 7841	. . . . . 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 17	. . . . 5 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) ∈ V)
19		simpl 483	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝑓:𝐴–1-1→𝐵)
20		f1dm 6742	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1→𝐵 → dom 𝑓 = 𝐴)
21	4	dmex 7848	. . . . . . . . . 10 ⊢ dom 𝑓 ∈ V
22	20, 21	eqeltrrdi 2847	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
23	22	adantr 481	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
24		simpr 485	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
25		eqid 2736	. . . . . . . . 9 ⊢ ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) = ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})))
26	25	domss2 9080	. . . . . . . 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 1371	. . . . . . 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 1143	. . . . . 6 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐴 ⊆ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))))
29	27	simp1d 1142	. . . . . . 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 8906	. . . . . . 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 584	. . . . . 6 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐵 ≈ ran ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))))
32	28, 31	jca 512	. . . . 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 3970	. . . . . 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 5109	. . . . . 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 631	. . . . 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 3557	. . . 4 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥))
37	36	ex 413	. . 3 ⊢ (𝑓:𝐴–1-1→𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥)))
38	37	exlimiv 1933	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥)))
39	1, 3, 38	sylc 65	1 ⊢ (𝐴 ≼ 𝐵 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3445 ∖ cdif 3907 ∪ cun 3908 ⊆ wss 3910 𝒫 cpw 4560 {csn 4586 ∪ cuni 4865 class class class wbr 5105 I cid 5530 × cxp 5631 ^◡ccnv 5632 dom cdm 5633 ran crn 5634 ↾ cres 5635 ∘ ccom 5637 ⟶wf 6492 –1-1→wf1 6493 –1-1-onto→wf1o 6495 1^st c1st 7919 ≈ cen 8880 ≼ cdom 8881

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-1st 7921 df-2nd 7922 df-en 8884 df-dom 8885

This theorem is referenced by: (None)

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