Theorem domdifsn 8043

Description: Dominance over a set with one element removed. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
domdifsn	⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ (𝐵 ∖ {𝐶}))

Proof of Theorem domdifsn

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sdomdom 7983	. . . . 5 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵)
2		relsdom 7962	. . . . . . 7 ⊢ Rel ≺
3	2	brrelex2i 5159	. . . . . 6 ⊢ (𝐴 ≺ 𝐵 → 𝐵 ∈ V)
4		brdomg 7965	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
5	3, 4	syl 17	. . . . 5 ⊢ (𝐴 ≺ 𝐵 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
6	1, 5	mpbid 222	. . . 4 ⊢ (𝐴 ≺ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
7	6	adantr 481	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
8		f1f 6101	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
9		frn 6053	. . . . . . . 8 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝑓:𝐴–1-1→𝐵 → ran 𝑓 ⊆ 𝐵)
11	10	adantl 482	. . . . . 6 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ⊆ 𝐵)
12		sdomnen 7984	. . . . . . . 8 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
13	12	ad2antrr 762	. . . . . . 7 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ¬ 𝐴 ≈ 𝐵)
14		vex 3203	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
15		dff1o5 6146	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1-onto→𝐵 ↔ (𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵))
16	15	biimpri 218	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵) → 𝑓:𝐴–1-1-onto→𝐵)
17		f1oen3g 7971	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
18	14, 16, 17	sylancr 695	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵) → 𝐴 ≈ 𝐵)
19	18	ex 450	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1→𝐵 → (ran 𝑓 = 𝐵 → 𝐴 ≈ 𝐵))
20	19	necon3bd 2808	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1→𝐵 → (¬ 𝐴 ≈ 𝐵 → ran 𝑓 ≠ 𝐵))
21	20	adantl 482	. . . . . . 7 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (¬ 𝐴 ≈ 𝐵 → ran 𝑓 ≠ 𝐵))
22	13, 21	mpd 15	. . . . . 6 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ≠ 𝐵)
23		pssdifn0 3944	. . . . . 6 ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ ran 𝑓 ≠ 𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅)
24	11, 22, 23	syl2anc 693	. . . . 5 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅)
25		n0 3931	. . . . 5 ⊢ ((𝐵 ∖ ran 𝑓) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓))
26	24, 25	sylib 208	. . . 4 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓))
27	2	brrelexi 5158	. . . . . . . . 9 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V)
28	27	ad2antrr 762	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ∈ V)
29	3	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐵 ∈ V)
30		difexg 4808	. . . . . . . . 9 ⊢ (𝐵 ∈ V → (𝐵 ∖ {𝑥}) ∈ V)
31	29, 30	syl 17	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ∈ V)
32		eldifn 3733	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → ¬ 𝑥 ∈ ran 𝑓)
33		disjsn 4246	. . . . . . . . . . . . 13 ⊢ ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝑓)
34	32, 33	sylibr 224	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → (ran 𝑓 ∩ {𝑥}) = ∅)
35	34	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → (ran 𝑓 ∩ {𝑥}) = ∅)
36	10	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓 ⊆ 𝐵)
37		reldisj 4020	. . . . . . . . . . . 12 ⊢ (ran 𝑓 ⊆ 𝐵 → ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})))
38	36, 37	syl 17	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})))
39	35, 38	mpbid 222	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓 ⊆ (𝐵 ∖ {𝑥}))
40		f1ssr 6107	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥}))
41	39, 40	syldan 487	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥}))
42	41	adantl 482	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥}))
43		f1dom2g 7973	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ (𝐵 ∖ {𝑥}) ∈ V ∧ 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥})) → 𝐴 ≼ (𝐵 ∖ {𝑥}))
44	28, 31, 42, 43	syl3anc 1326	. . . . . . 7 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝑥}))
45		eldifi 3732	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝑥 ∈ 𝐵)
46	45	ad2antll 765	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑥 ∈ 𝐵)
47		simplr 792	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐶 ∈ 𝐵)
48		difsnen 8042	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶}))
49	29, 46, 47, 48	syl3anc 1326	. . . . . . 7 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶}))
50		domentr 8015	. . . . . . 7 ⊢ ((𝐴 ≼ (𝐵 ∖ {𝑥}) ∧ (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
51	44, 49, 50	syl2anc 693	. . . . . 6 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
52	51	expr 643	. . . . 5 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝐴 ≼ (𝐵 ∖ {𝐶})))
53	52	exlimdv 1861	. . . 4 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝐴 ≼ (𝐵 ∖ {𝐶})))
54	26, 53	mpd 15	. . 3 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
55	7, 54	exlimddv 1863	. 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
56	1	adantr 481	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → 𝐴 ≼ 𝐵)
57		difsn 4328	. . . . 5 ⊢ (¬ 𝐶 ∈ 𝐵 → (𝐵 ∖ {𝐶}) = 𝐵)
58	57	breq2d 4665	. . . 4 ⊢ (¬ 𝐶 ∈ 𝐵 → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴 ≼ 𝐵))
59	58	adantl 482	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴 ≼ 𝐵))
60	56, 59	mpbird 247	. 2 ⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
61	55, 60	pm2.61dan 832	1 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ (𝐵 ∖ {𝐶}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653  ran crn 5115  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by:  domunsn  8110  marypha1lem  8339