Theorem fv2ndcnv 36309

Description: The value of the converse of 2^nd restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)

Assertion

Ref	Expression
fv2ndcnv	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩)

Proof of Theorem fv2ndcnv

Step	Hyp	Ref	Expression
1		snidg 4628	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
2	1	anim1i 627	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴))
3		eqid 2765	. . 3 ⊢ 𝑌 = 𝑌
4	2, 3	jctir 530	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌))
5		2ndconst 8102	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto→𝐴)
6	5	adantr 486	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto→𝐴)
7		f1ocnv 6837	. . . . 5 ⊢ ((2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto→𝐴 → ◡(2nd ↾ ({𝑋} × 𝐴)):𝐴–1-1-onto→({𝑋} × 𝐴))
8		f1ofn 6825	. . . . 5 ⊢ (◡(2nd ↾ ({𝑋} × 𝐴)):𝐴–1-1-onto→({𝑋} × 𝐴) → ◡(2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴)
9	6, 7, 8	3syl 19	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ◡(2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴)
10		fnbrfvb 6935	. . . 4 ⊢ ((◡(2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴 ∧ 𝑌 ∈ 𝐴) → ((◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
11	9, 10	sylancom 600	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
12		opex 5447	. . . . . 6 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
13		brcnvg 5867	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
14	12, 13	mpan2 704	. . . . 5 ⊢ (𝑌 ∈ 𝐴 → (𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
15	14	adantl 487	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
16		brres 5987	. . . . . 6 ⊢ (𝑌 ∈ 𝐴 → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌)))
17	16	adantl 487	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌)))
18		opelxp 5699	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ↔ (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴))
19	18	anbi1i 636	. . . . . 6 ⊢ ((⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌))
20		br2ndeqg 8015	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (⟨𝑋, 𝑌⟩2nd 𝑌 ↔ 𝑌 = 𝑌))
21	20	anbi2d 642	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌)))
22	19, 21	bitrid 286	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌)))
23	17, 22	bitrd 282	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌)))
24	15, 23	bitrd 282	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌)))
25	11, 24	bitrd 282	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌)))
26	4, 25	mpbird 260	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  {csn 4591  ⟨cop 4597   class class class wbr 5111   × cxp 5661  ^◡ccnv 5662   ↾ cres 5665   Fn wfn 6535  –1-1-onto→wf1o 6539  ‘cfv 6540  2^nd c2nd 7991

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-1st 7992  df-2nd 7993

This theorem is used by: (None)