Theorem 2ndconst 7153

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

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 4251	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
2		fo2ndres 7084	. . 3 ⊢ ({𝐴} ≠ ∅ → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto→𝐵)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto→𝐵)
4		moeq 3349	. . . . . 6 ⊢ ∃*𝑥 𝑥 = ⟨𝐴, 𝑦⟩
5	4	moani 2513	. . . . 5 ⊢ ∃*𝑥(𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)
6		vex 3176	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	6	brres 5323	. . . . . . 7 ⊢ (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥2nd 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)))
8		fo2nd 7080	. . . . . . . . . . 11 ⊢ 2nd :V–onto→V
9		fofn 6030	. . . . . . . . . . 11 ⊢ (2nd :V–onto→V → 2nd Fn V)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ 2nd Fn V
11		vex 3176	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		fnbrfvb 6146	. . . . . . . . . 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	anbi1i 727	. . . . . . . 8 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑥2nd 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)))
15		elxp7 7092	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵)))
16		eleq1 2676	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑥) = 𝑦 → ((2nd ‘𝑥) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
17	16	biimpa 500	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (2nd ‘𝑥) ∈ 𝐵) → 𝑦 ∈ 𝐵)
18	17	adantrl 748	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵)) → 𝑦 ∈ 𝐵)
19	18	adantrl 748	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵))) → 𝑦 ∈ 𝐵)
20		elsni 4142	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑥) ∈ {𝐴} → (1st ‘𝑥) = 𝐴)
21		eqopi 7093	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) = 𝐴 ∧ (2nd ‘𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
22	21	ancom2s 840	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (V × V) ∧ ((2nd ‘𝑥) = 𝑦 ∧ (1st ‘𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
23	22	an12s 839	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st ‘𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
24	20, 23	sylanr2 683	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st ‘𝑥) ∈ {𝐴})) → 𝑥 = ⟨𝐴, 𝑦⟩)
25	24	adantrrr 757	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵))) → 𝑥 = ⟨𝐴, 𝑦⟩)
26	19, 25	jca 553	. . . . . . . . . . 11 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ {𝐴} ∧ (2nd ‘𝑥) ∈ 𝐵))) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩))
27	15, 26	sylan2b 491	. . . . . . . . . 10 ⊢ (((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩))
28	27	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵))) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩))
29		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
30		op2ndg 7072	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
31	6, 30	mpan2 703	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
32	29, 31	sylan9eqr 2666	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = ⟨𝐴, 𝑦⟩) → (2nd ‘𝑥) = 𝑦)
33	32	adantrl 748	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd ‘𝑥) = 𝑦)
34		simprr 792	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
35		snidg 4153	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
36	35	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
37		simprl 790	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦 ∈ 𝐵)
38		opelxpi 5072	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
39	36, 37, 38	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
40	34, 39	eqeltrd 2688	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
41	33, 40	jca 553	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)) → ((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)))
42	28, 41	impbida 873	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (((2nd ‘𝑥) = 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
43	14, 42	syl5bbr 273	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((𝑥2nd 𝑦 ∧ 𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
44	7, 43	syl5bb 271	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
45	44	mobidv 2479	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐵 ∧ 𝑥 = ⟨𝐴, 𝑦⟩)))
46	5, 45	mpbiri 247	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
47	46	alrimiv 1842	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
48		funcnv2 5871	. . 3 ⊢ (Fun ◡(2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
49	47, 48	sylibr 223	. 2 ⊢ (𝐴 ∈ 𝑉 → Fun ◡(2nd ↾ ({𝐴} × 𝐵)))
50		dff1o3 6056	. 2 ⊢ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵 ↔ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto→𝐵 ∧ Fun ◡(2nd ↾ ({𝐴} × 𝐵))))
51	3, 49, 50	sylanbrc 695	1 ⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃*wmo 2459   ≠ wne 2780  Vcvv 3173  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ^◡ccnv 5037   ↾ cres 5040  Fun wfun 5798   Fn wfn 5799  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  1^st c1st 7057  2^nd c2nd 7058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1st 7059  df-2nd 7060

This theorem is referenced by:  curry1  7156  xpfi  8116  fsum2dlem  14343  fprod2dlem  14549  gsum2dlem2  18193  ovoliunlem1  23077  gsummpt2d  29112  fv2ndcnv  30926