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Theorem domssex 9108

Description: Weakening of domssex2 9107 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 8934	. 2 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
2		reldom 8927	. . 3 ⊢ Rel ≼
3	2	brrelex2i 5698	. 2 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
4		vex 3454	. . . . . . . 8 ⊢ 𝑓 ∈ V
5		f1stres 7995	. . . . . . . . 9 ⊢ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})⟶(𝐵 ∖ ran 𝑓)
6		difexg 5287	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
7	6	adantl 481	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → (𝐵 ∖ ran 𝑓) ∈ V)
8		snex 5394	. . . . . . . . . 10 ⊢ {𝒫 ∪ ran 𝐴} ∈ V
9		xpexg 7729	. . . . . . . . . 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 7915	. . . . . . . . 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 1468	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})) ∈ V)
13		unexg 7722	. . . . . . . 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 7903	. . . . . . 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 7881	. . . . . 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 482	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝑓:𝐴–1-1→𝐵)
20		f1dm 6763	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1→𝐵 → dom 𝑓 = 𝐴)
21	4	dmex 7888	. . . . . . . . . 10 ⊢ dom 𝑓 ∈ V
22	20, 21	eqeltrrdi 2838	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
23	22	adantr 480	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
24		simpr 484	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
25		eqid 2730	. . . . . . . . 9 ⊢ ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴}))) = ^◡(𝑓 ∪ (1^st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ∪ ran 𝐴})))
26	25	domss2 9106	. . . . . . . 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 1373	. . . . . . 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 8941	. . . . . . 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 511	. . . . 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 3976	. . . . . 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 5114	. . . . . 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 632	. . . . 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 3567	. . . 4 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥))
37	36	ex 412	. . 3 ⊢ (𝑓:𝐴–1-1→𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥)))
38	37	exlimiv 1930	. 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 395 ∧ w3a 1086 = wceq 1540 ∃wex 1779 ∈ wcel 2109 Vcvv 3450 ∖ cdif 3914 ∪ cun 3915 ⊆ wss 3917 𝒫 cpw 4566 {csn 4592 ∪ cuni 4874 class class class wbr 5110 I cid 5535 × cxp 5639 ^◡ccnv 5640 dom cdm 5641 ran crn 5642 ↾ cres 5643 ∘ ccom 5645 ⟶wf 6510 –1-1→wf1 6511 –1-1-onto→wf1o 6513 1^st c1st 7969 ≈ cen 8918 ≼ cdom 8919

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-1st 7971 df-2nd 7972 df-en 8922 df-dom 8923

This theorem is referenced by: (None)

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