Theorem domdifsn 9120

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 9040	. . . . 5 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵)
2		relsdom 9010	. . . . . . 7 ⊢ Rel ≺
3	2	brrelex2i 5757	. . . . . 6 ⊢ (𝐴 ≺ 𝐵 → 𝐵 ∈ V)
4		brdomg 9016	. . . . . 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 232	. . . 4 ⊢ (𝐴 ≺ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
7	6	adantr 480	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
8		f1f 6817	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
9	8	frnd 6755	. . . . . . 7 ⊢ (𝑓:𝐴–1-1→𝐵 → ran 𝑓 ⊆ 𝐵)
10	9	adantl 481	. . . . . 6 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ⊆ 𝐵)
11		sdomnen 9041	. . . . . . . 8 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
12	11	ad2antrr 725	. . . . . . 7 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ¬ 𝐴 ≈ 𝐵)
13		vex 3492	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
14		dff1o5 6871	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1-onto→𝐵 ↔ (𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵))
15	14	biimpri 228	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵) → 𝑓:𝐴–1-1-onto→𝐵)
16		f1oen3g 9026	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
17	13, 15, 16	sylancr 586	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵) → 𝐴 ≈ 𝐵)
18	17	ex 412	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1→𝐵 → (ran 𝑓 = 𝐵 → 𝐴 ≈ 𝐵))
19	18	necon3bd 2960	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1→𝐵 → (¬ 𝐴 ≈ 𝐵 → ran 𝑓 ≠ 𝐵))
20	19	adantl 481	. . . . . . 7 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (¬ 𝐴 ≈ 𝐵 → ran 𝑓 ≠ 𝐵))
21	12, 20	mpd 15	. . . . . 6 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ≠ 𝐵)
22		pssdifn0 4391	. . . . . 6 ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ ran 𝑓 ≠ 𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅)
23	10, 21, 22	syl2anc 583	. . . . 5 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅)
24		n0 4376	. . . . 5 ⊢ ((𝐵 ∖ ran 𝑓) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓))
25	23, 24	sylib 218	. . . 4 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓))
26	2	brrelex1i 5756	. . . . . . . . 9 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V)
27	26	ad2antrr 725	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ∈ V)
28	3	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐵 ∈ V)
29	28	difexd 5349	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ∈ V)
30		eldifn 4155	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → ¬ 𝑥 ∈ ran 𝑓)
31		disjsn 4736	. . . . . . . . . . . . 13 ⊢ ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝑓)
32	30, 31	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → (ran 𝑓 ∩ {𝑥}) = ∅)
33	32	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → (ran 𝑓 ∩ {𝑥}) = ∅)
34	9	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓 ⊆ 𝐵)
35		reldisj 4476	. . . . . . . . . . . 12 ⊢ (ran 𝑓 ⊆ 𝐵 → ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})))
36	34, 35	syl 17	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})))
37	33, 36	mpbid 232	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓 ⊆ (𝐵 ∖ {𝑥}))
38		f1ssr 6823	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥}))
39	37, 38	syldan 590	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥}))
40	39	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥}))
41		f1dom2g 9029	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ (𝐵 ∖ {𝑥}) ∈ V ∧ 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥})) → 𝐴 ≼ (𝐵 ∖ {𝑥}))
42	27, 29, 40, 41	syl3anc 1371	. . . . . . 7 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝑥}))
43		eldifi 4154	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝑥 ∈ 𝐵)
44	43	ad2antll 728	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑥 ∈ 𝐵)
45		simplr 768	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐶 ∈ 𝐵)
46		difsnen 9119	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶}))
47	28, 44, 45, 46	syl3anc 1371	. . . . . . 7 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶}))
48		domentr 9073	. . . . . . 7 ⊢ ((𝐴 ≼ (𝐵 ∖ {𝑥}) ∧ (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
49	42, 47, 48	syl2anc 583	. . . . . 6 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
50	49	expr 456	. . . . 5 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝐴 ≼ (𝐵 ∖ {𝐶})))
51	50	exlimdv 1932	. . . 4 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝐴 ≼ (𝐵 ∖ {𝐶})))
52	25, 51	mpd 15	. . 3 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
53	7, 52	exlimddv 1934	. 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
54	1	adantr 480	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → 𝐴 ≼ 𝐵)
55		difsn 4823	. . . . 5 ⊢ (¬ 𝐶 ∈ 𝐵 → (𝐵 ∖ {𝐶}) = 𝐵)
56	55	breq2d 5178	. . . 4 ⊢ (¬ 𝐶 ∈ 𝐵 → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴 ≼ 𝐵))
57	56	adantl 481	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴 ≼ 𝐵))
58	54, 57	mpbird 257	. 2 ⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
59	53, 58	pm2.61dan 812	1 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ (𝐵 ∖ {𝐶}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648   class class class wbr 5166  ran crn 5701  –1-1→wf1 6570  –1-1-onto→wf1o 6572   ≈ cen 9000   ≼ cdom 9001   ≺ csdm 9002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-en 9004  df-dom 9005  df-sdom 9006

This theorem is referenced by:  domunsn  9193  marypha1lem  9502