Theorem domdifsn 9093

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 9019	. . . . 5 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵)
2		relsdom 8991	. . . . . . 7 ⊢ Rel ≺
3	2	brrelex2i 5746	. . . . . 6 ⊢ (𝐴 ≺ 𝐵 → 𝐵 ∈ V)
4		brdomg 8996	. . . . . 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 6805	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
9	8	frnd 6745	. . . . . . 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 9020	. . . . . . . 8 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
12	11	ad2antrr 726	. . . . . . 7 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ¬ 𝐴 ≈ 𝐵)
13		vex 3482	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
14		dff1o5 6858	. . . . . . . . . . . 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 9006	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
17	13, 15, 16	sylancr 587	. . . . . . . . . 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 2952	. . . . . . . 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 4374	. . . . . 6 ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ ran 𝑓 ≠ 𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅)
23	10, 21, 22	syl2anc 584	. . . . 5 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅)
24		n0 4359	. . . . 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 5745	. . . . . . . . 9 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V)
27	26	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ∈ V)
28	3	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐵 ∈ V)
29	28	difexd 5337	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ∈ V)
30		eldifn 4142	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → ¬ 𝑥 ∈ ran 𝑓)
31		disjsn 4716	. . . . . . . . . . . . 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 4459	. . . . . . . . . . . 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 6811	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥}))
39	37, 38	syldan 591	. . . . . . . . 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 9009	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ (𝐵 ∖ {𝑥}) ∈ V ∧ 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥})) → 𝐴 ≼ (𝐵 ∖ {𝑥}))
42	27, 29, 40, 41	syl3anc 1370	. . . . . . 7 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝑥}))
43		eldifi 4141	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝑥 ∈ 𝐵)
44	43	ad2antll 729	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑥 ∈ 𝐵)
45		simplr 769	. . . . . . . 8 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐶 ∈ 𝐵)
46		difsnen 9092	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶}))
47	28, 44, 45, 46	syl3anc 1370	. . . . . . 7 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶}))
48		domentr 9052	. . . . . . 7 ⊢ ((𝐴 ≼ (𝐵 ∖ {𝑥}) ∧ (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
49	42, 47, 48	syl2anc 584	. . . . . 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 1931	. . . 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 1933	. 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
54	1	adantr 480	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → 𝐴 ≼ 𝐵)
55		difsn 4803	. . . . 5 ⊢ (¬ 𝐶 ∈ 𝐵 → (𝐵 ∖ {𝐶}) = 𝐵)
56	55	breq2d 5160	. . . 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 813	1 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ (𝐵 ∖ {𝐶}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  Vcvv 3478   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  {csn 4631   class class class wbr 5148  ran crn 5690  –1-1→wf1 6560  –1-1-onto→wf1o 6562   ≈ cen 8981   ≼ cdom 8982   ≺ csdm 8983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-en 8985  df-dom 8986  df-sdom 8987

This theorem is referenced by:  domunsn  9166  marypha1lem  9471