Theorem 1stconst 7265

Description: The mapping of a restriction of the 1^st function to a constant function. (Contributed by NM, 14-Dec-2008.)

Assertion

Ref	Expression
1stconst	⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴)

Proof of Theorem 1stconst

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

Step	Hyp	Ref	Expression
1		snnzg 4308	. . 3 ⊢ (𝐵 ∈ 𝑉 → {𝐵} ≠ ∅)
2		fo1stres 7192	. . 3 ⊢ ({𝐵} ≠ ∅ → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴)
3	1, 2	syl 17	. 2 ⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴)
4		moeq 3382	. . . . . 6 ⊢ ∃*𝑥 𝑥 = ⟨𝑦, 𝐵⟩
5	4	moani 2525	. . . . 5 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)
6		vex 3203	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	6	brres 5402	. . . . . . 7 ⊢ (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥1st 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})))
8		fo1st 7188	. . . . . . . . . . 11 ⊢ 1st :V–onto→V
9		fofn 6117	. . . . . . . . . . 11 ⊢ (1st :V–onto→V → 1st Fn V)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ 1st Fn V
11		vex 3203	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		fnbrfvb 6236	. . . . . . . . . 10 ⊢ ((1st Fn V ∧ 𝑥 ∈ V) → ((1st ‘𝑥) = 𝑦 ↔ 𝑥1st 𝑦))
13	10, 11, 12	mp2an 708	. . . . . . . . 9 ⊢ ((1st ‘𝑥) = 𝑦 ↔ 𝑥1st 𝑦)
14	13	anbi1i 731	. . . . . . . 8 ⊢ (((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑥1st 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})))
15		elxp7 7201	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})))
16		eleq1 2689	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑥) = 𝑦 → ((1st ‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17	16	biimpa 501	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (1st ‘𝑥) ∈ 𝐴) → 𝑦 ∈ 𝐴)
18	17	adantrr 753	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑥) = 𝑦 ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})) → 𝑦 ∈ 𝐴)
19	18	adantrl 752	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵}))) → 𝑦 ∈ 𝐴)
20		elsni 4194	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑥) ∈ {𝐵} → (2nd ‘𝑥) = 𝐵)
21		eqopi 7202	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) = 𝑦 ∧ (2nd ‘𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
22	21	an12s 843	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd ‘𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
23	20, 22	sylanr2 685	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd ‘𝑥) ∈ {𝐵})) → 𝑥 = ⟨𝑦, 𝐵⟩)
24	23	adantrrl 760	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵}))) → 𝑥 = ⟨𝑦, 𝐵⟩)
25	19, 24	jca 554	. . . . . . . . . . 11 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵}))) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
26	15, 25	sylan2b 492	. . . . . . . . . 10 ⊢ (((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
27	26	adantl 482	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ ((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵}))) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
28		simprr 796	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
29	28	fveq2d 6195	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
30		simprl 794	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦 ∈ 𝐴)
31		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ 𝑉)
32		op1stg 7180	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
33	30, 31, 32	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
34	29, 33	eqtrd 2656	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘𝑥) = 𝑦)
35		snidg 4206	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
36	35	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
37		opelxpi 5148	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐵 ∈ {𝐵}) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
38	30, 36, 37	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
39	28, 38	eqeltrd 2701	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
40	34, 39	jca 554	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → ((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})))
41	27, 40	impbida 877	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
42	14, 41	syl5bbr 274	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ((𝑥1st 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
43	7, 42	syl5bb 272	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
44	43	mobidv 2491	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
45	5, 44	mpbiri 248	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
46	45	alrimiv 1855	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
47		funcnv2 5957	. . 3 ⊢ (Fun ◡(1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
48	46, 47	sylibr 224	. 2 ⊢ (𝐵 ∈ 𝑉 → Fun ◡(1st ↾ (𝐴 × {𝐵})))
49		dff1o3 6143	. 2 ⊢ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴 ↔ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴 ∧ Fun ◡(1st ↾ (𝐴 × {𝐵}))))
50	3, 48, 49	sylanbrc 698	1 ⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∃*wmo 2471   ≠ wne 2794  Vcvv 3200  ∅c0 3915  {csn 4177  ⟨cop 4183   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113   ↾ cres 5116  Fun wfun 5882   Fn wfn 5883  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  1^st c1st 7166  2^nd c2nd 7167

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-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  curry2  7272  domss2  8119  fv1stcnv  31680