Theorem fv2ndcnv 33339

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 4560	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
2	1	anim1i 618	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴))
3		eqid 2739	. . 3 ⊢ 𝑌 = 𝑌
4	2, 3	jctir 524	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌))
5		2ndconst 7835	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto→𝐴)
6	5	adantr 484	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto→𝐴)
7		f1ocnv 6643	. . . . 5 ⊢ ((2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto→𝐴 → ◡(2nd ↾ ({𝑋} × 𝐴)):𝐴–1-1-onto→({𝑋} × 𝐴))
8		f1ofn 6632	. . . . 5 ⊢ (◡(2nd ↾ ({𝑋} × 𝐴)):𝐴–1-1-onto→({𝑋} × 𝐴) → ◡(2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴)
9	6, 7, 8	3syl 18	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ◡(2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴)
10		fnbrfvb 6735	. . . 4 ⊢ ((◡(2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴 ∧ 𝑌 ∈ 𝐴) → ((◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
11	9, 10	sylancom 591	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
12		opex 5332	. . . . . 6 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
13		brcnvg 5732	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
14	12, 13	mpan2 691	. . . . 5 ⊢ (𝑌 ∈ 𝐴 → (𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
15	14	adantl 485	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
16		brres 5842	. . . . . 6 ⊢ (𝑌 ∈ 𝐴 → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌)))
17	16	adantl 485	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌)))
18		opelxp 5571	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ↔ (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴))
19	18	anbi1i 627	. . . . . 6 ⊢ ((⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌))
20		br2ndeqg 7750	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (⟨𝑋, 𝑌⟩2nd 𝑌 ↔ 𝑌 = 𝑌))
21	20	anbi2d 632	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌)))
22	19, 21	syl5bb 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⟩)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2114  Vcvv 3400  {csn 4526  ⟨cop 4532   class class class wbr 5040   × cxp 5533  ^◡ccnv 5534   ↾ cres 5537   Fn wfn 6345  –1-1-onto→wf1o 6349  ‘cfv 6350  2^nd c2nd 7726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7492

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-1st 7727  df-2nd 7728

This theorem is referenced by: (None)