Theorem 1stconst 6451

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		snmg 3829	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐵})
2		fo1stresm 6389	. . 3 ⊢ (∃𝑥 𝑥 ∈ {𝐵} → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴)
3	1, 2	syl 14	. 2 ⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴)
4		moeq 3001	. . . . . 6 ⊢ ∃*𝑥 𝑥 = ⟨𝑦, 𝐵⟩
5	4	moani 2157	. . . . 5 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)
6		vex 2824	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	6	brres 5067	. . . . . . 7 ⊢ (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥1st 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})))
8		fo1st 6385	. . . . . . . . . . 11 ⊢ 1st :V–onto→V
9		fofn 5615	. . . . . . . . . . 11 ⊢ (1st :V–onto→V → 1st Fn V)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ 1st Fn V
11		vex 2824	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		fnbrfvb 5738	. . . . . . . . . 10 ⊢ ((1st Fn V ∧ 𝑥 ∈ V) → ((1st ‘𝑥) = 𝑦 ↔ 𝑥1st 𝑦))
13	10, 11, 12	mp2an 430	. . . . . . . . 9 ⊢ ((1st ‘𝑥) = 𝑦 ↔ 𝑥1st 𝑦)
14	13	anbi1i 462	. . . . . . . 8 ⊢ (((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑥1st 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})))
15		elxp7 6398	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})))
16		eleq1 2301	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑥) = 𝑦 → ((1st ‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17	16	biimpa 296	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (1st ‘𝑥) ∈ 𝐴) → 𝑦 ∈ 𝐴)
18	17	adantrr 483	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑥) = 𝑦 ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})) → 𝑦 ∈ 𝐴)
19	18	adantrl 482	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵}))) → 𝑦 ∈ 𝐴)
20		elsni 3726	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑥) ∈ {𝐵} → (2nd ‘𝑥) = 𝐵)
21		eqopi 6400	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) = 𝑦 ∧ (2nd ‘𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
22	21	an12s 571	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd ‘𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
23	20, 22	sylanr2 409	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd ‘𝑥) ∈ {𝐵})) → 𝑥 = ⟨𝑦, 𝐵⟩)
24	23	adantrrl 490	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵}))) → 𝑥 = ⟨𝑦, 𝐵⟩)
25	19, 24	jca 306	. . . . . . . . . . 11 ⊢ (((1st ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵}))) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
26	15, 25	sylan2b 287	. . . . . . . . . 10 ⊢ (((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
27	26	adantl 277	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ ((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵}))) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
28		simprr 537	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
29	28	fveq2d 5697	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
30		simprl 535	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦 ∈ 𝐴)
31		simpl 109	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ 𝑉)
32		op1stg 6378	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
33	30, 31, 32	syl2anc 415	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
34	29, 33	eqtrd 2271	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘𝑥) = 𝑦)
35		snidg 3737	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
36	35	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
37		opelxpi 4804	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐵 ∈ {𝐵}) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
38	30, 36, 37	syl2anc 415	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
39	28, 38	eqeltrd 2315	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
40	34, 39	jca 306	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → ((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})))
41	27, 40	impbida 604	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (((1st ‘𝑥) = 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
42	14, 41	bitr3id 194	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ((𝑥1st 𝑦 ∧ 𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
43	7, 42	bitrid 192	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
44	43	mobidv 2122	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
45	5, 44	mpbiri 168	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
46	45	alrimiv 1927	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
47		funcnv2 5439	. . 3 ⊢ (Fun ◡(1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
48	46, 47	sylibr 134	. 2 ⊢ (𝐵 ∈ 𝑉 → Fun ◡(1st ↾ (𝐴 × {𝐵})))
49		dff1o3 5643	. 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 421	1 ⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1400   = wceq 1402  ∃wex 1545  ∃*wmo 2087   ∈ wcel 2209  Vcvv 2821  {csn 3708  ⟨cop 3711   class class class wbr 4128   × cxp 4770  ^◡ccnv 4771   ↾ cres 4774  Fun wfun 5369   Fn wfn 5370  –onto→wfo 5373  –1-1-onto→wf1o 5374  ‘cfv 5375  1^st c1st 6366  2^nd c2nd 6367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1st 6368  df-2nd 6369

This theorem is referenced by: (None)