Theorem 1stconst 8125

Description: The mapping of a restriction of the 1^st function to a constant function. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

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 4774	. . 3 ⊢ (𝐵 ∈ 𝑉 → {𝐵} ≠ ∅)
2		fo1stres 8040	. . 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 3713	. . . . . 6 ⊢ ∃*𝑥 𝑥 = ⟨𝑦, 𝐵⟩
5	4	moani 2553	. . . . 5 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)
6		vex 3484	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	6	brresi 6006	. . . . . . 7 ⊢ (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦))
8		fo1st 8034	. . . . . . . . . . 11 ⊢ 1st :V–onto→V
9		fofn 6822	. . . . . . . . . . 11 ⊢ (1st :V–onto→V → 1st Fn V)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ 1st Fn V
11		vex 3484	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		fnbrfvb 6959	. . . . . . . . . 10 ⊢ ((1st Fn V ∧ 𝑥 ∈ V) → ((1st ‘𝑥) = 𝑦 ↔ 𝑥1st 𝑦))
13	10, 11, 12	mp2an 692	. . . . . . . . 9 ⊢ ((1st ‘𝑥) = 𝑦 ↔ 𝑥1st 𝑦)
14	13	anbi2i 623	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st ‘𝑥) = 𝑦) ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦))
15		elxp7 8049	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})))
16		eleq1 2829	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑥) = 𝑦 → ((1st ‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17	16	biimpac 478	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑥) ∈ 𝐴 ∧ (1st ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴)
18	17	adantlr 715	. . . . . . . . . . . . 13 ⊢ ((((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵}) ∧ (1st ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴)
19	18	adantll 714	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})) ∧ (1st ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴)
20		elsni 4643	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑥) ∈ {𝐵} → (2nd ‘𝑥) = 𝐵)
21		eqopi 8050	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) = 𝑦 ∧ (2nd ‘𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
22	21	anass1rs 655	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (V × V) ∧ (2nd ‘𝑥) = 𝐵) ∧ (1st ‘𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
23	20, 22	sylanl2 681	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (V × V) ∧ (2nd ‘𝑥) ∈ {𝐵}) ∧ (1st ‘𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
24	23	adantlrl 720	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})) ∧ (1st ‘𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
25	19, 24	jca 511	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})) ∧ (1st ‘𝑥) = 𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
26	15, 25	sylanb 581	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st ‘𝑥) = 𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
27	26	adantl 481	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st ‘𝑥) = 𝑦)) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
28		simprr 773	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
29		simprl 771	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦 ∈ 𝐴)
30		snidg 4660	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
31	30	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
32	29, 31	opelxpd 5724	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
33	28, 32	eqeltrd 2841	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
34	28	fveq2d 6910	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
35		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ 𝑉)
36		op1stg 8026	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
37	29, 35, 36	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
38	34, 37	eqtrd 2777	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘𝑥) = 𝑦)
39	33, 38	jca 511	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st ‘𝑥) = 𝑦))
40	27, 39	impbida 801	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st ‘𝑥) = 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
41	14, 40	bitr3id 285	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
42	7, 41	bitrid 283	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
43	42	mobidv 2549	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
44	5, 43	mpbiri 258	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
45	44	alrimiv 1927	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
46		funcnv2 6634	. . 3 ⊢ (Fun ◡(1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
47	45, 46	sylibr 234	. 2 ⊢ (𝐵 ∈ 𝑉 → Fun ◡(1st ↾ (𝐴 × {𝐵})))
48		dff1o3 6854	. 2 ⊢ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴 ↔ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴 ∧ Fun ◡(1st ↾ (𝐴 × {𝐵}))))
49	3, 47, 48	sylanbrc 583	1 ⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2108  ∃*wmo 2538   ≠ wne 2940  Vcvv 3480  ∅c0 4333  {csn 4626  ⟨cop 4632   class class class wbr 5143   × cxp 5683  ^◡ccnv 5684   ↾ cres 5687  Fun wfun 6555   Fn wfn 6556  –onto→wfo 6559  –1-1-onto→wf1o 6560  ‘cfv 6561  1^st c1st 8012  2^nd c2nd 8013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015

This theorem is referenced by:  curry2  8132  domss2  9176  fv1stcnv  35777