Theorem 2ndconst 8096

Description: The mapping of a restriction of the 2^nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

Assertion

Ref	Expression
2ndconst	⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵)

Proof of Theorem 2ndconst

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

Step	Hyp	Ref	Expression
1		snnzg 4745	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
2		fo2ndres 8013	. . 3 ⊢ ({𝐴} ≠ ∅ → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto→𝐵)
3	1, 2	syl 18	. 2 ⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto→𝐵)
4		moeq 3679	. . . . . 6 ⊢ ∃*𝑥 𝑥 = ⟨𝐴, 𝑦⟩
5	4	moani 2587	. . . . 5 ⊢ ∃*𝑥(𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)
6		vex 3467	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	6	brresi 5988	. . . . . . 7 ⊢ (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦))
8		fo2nd 8007	. . . . . . . . . . 11 ⊢ 2nd :V–onto→V
9		fofn 6795	. . . . . . . . . . 11 ⊢ (2nd :V–onto→V → 2nd Fn V)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ 2nd Fn V
11		vex 3467	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		fnbrfvb 6932	. . . . . . . . . 10 ⊢ ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd ‘𝑥) = 𝑦 ↔ 𝑥2nd 𝑦))
13	10, 11, 12	mp2an 704	. . . . . . . . 9 ⊢ ((2nd ‘𝑥) = 𝑦 ↔ 𝑥2nd 𝑦)
14	13	anbi2i 634	. . . . . . . 8 ⊢ ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd ‘𝑥) = 𝑦) ↔ (𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦))
15		elxp7 8021	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵)))
16		eleq1 2857	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑥) = 𝑦 → ((2nd ‘𝑥) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
17	16	biimpac 483	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘𝑥) ∈ 𝐵 ∧ (2nd ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
18	17	adantll 726	. . . . . . . . . . . . 13 ⊢ ((((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵) ∧ (2nd ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
19	18	adantll 726	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵)) ∧ (2nd ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
20		elsni 4611	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑥) ∈ {𝐴} → (1st ‘𝑥) = 𝐴)
21		eqopi 8022	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) = 𝐴 ∧ (2nd ‘𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
22	21	anassrs 472	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (V × V) ∧ (1st ‘𝑥) = 𝐴) ∧ (2nd ‘𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
23	20, 22	sylanl2 693	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (V × V) ∧ (1st ‘𝑥) ∈ {𝐴}) ∧ (2nd ‘𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
24	23	adantlrr 733	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵)) ∧ (2nd ‘𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
25	19, 24	jca 520	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵)) ∧ (2nd ‘𝑥) = 𝑦) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩))
26	15, 25	sylanb 592	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd ‘𝑥) = 𝑦) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩))
27	26	adantl 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd ‘𝑥) = 𝑦)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩))
28		simprr 784	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
29		snidg 4631	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
30	29	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
31		simprl 782	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦 ∈ 𝐵)
32	30, 31	opelxpd 5701	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
33	28, 32	eqeltrd 2869	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
34		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
35		op2ndg 7999	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
36	35	elvd 3469	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
37	34, 36	sylan9eqr 2826	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = ⟨𝐴, 𝑦⟩) → (2nd ‘𝑥) = 𝑦)
38	37	adantrl 728	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd ‘𝑥) = 𝑦)
39	33, 38	jca 520	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → (𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd ‘𝑥) = 𝑦))
40	27, 39	impbida 812	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd ‘𝑥) = 𝑦) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
41	14, 40	bitr3id 288	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
42	7, 41	bitrid 286	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
43	42	mobidv 2583	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
44	5, 43	mpbiri 261	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
45	44	alrimiv 1954	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
46		funcnv2 6605	. . 3 ⊢ (Fun ◡(2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
47	45, 46	sylibr 237	. 2 ⊢ (𝐴 ∈ 𝑉 → Fun ◡(2nd ↾ ({𝐴} × 𝐵)))
48		dff1o3 6828	. 2 ⊢ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵 ↔ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto→𝐵 ∧ Fun ◡(2nd ↾ ({𝐴} × 𝐵))))
49	3, 47, 48	sylanbrc 594	1 ⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567   ∈ wcel 2149  ∃*wmo 2571   ≠ wne 2964  Vcvv 3463  ∅c0 4294  {csn 4594  ⟨cop 4600   class class class wbr 5113   × cxp 5660  ^◡ccnv 5661   ↾ cres 5664  Fun wfun 6531   Fn wfn 6532  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  1^st c1st 7984  2^nd c2nd 7985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1st 7986  df-2nd 7987

This theorem is referenced by:  curry1  8099  fsum2dlem  15821  fprod2dlem  16034  gsum2dlem2  20041  ovoliunlem1  25630  gsummpt2d  33310  fv2ndcnv  36203