Theorem fv2ndcnv 35758

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 4664	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
2	1	anim1i 615	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴))
3		eqid 2734	. . 3 ⊢ 𝑌 = 𝑌
4	2, 3	jctir 520	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌))
5		2ndconst 8124	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto→𝐴)
6	5	adantr 480	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto→𝐴)
7		f1ocnv 6860	. . . . 5 ⊢ ((2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto→𝐴 → ◡(2nd ↾ ({𝑋} × 𝐴)):𝐴–1-1-onto→({𝑋} × 𝐴))
8		f1ofn 6849	. . . . 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 6959	. . . 4 ⊢ ((◡(2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴 ∧ 𝑌 ∈ 𝐴) → ((◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
11	9, 10	sylancom 588	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
12		opex 5474	. . . . . 6 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
13		brcnvg 5892	. . . . . 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 481	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
16		brres 6006	. . . . . 6 ⊢ (𝑌 ∈ 𝐴 → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌)))
17	16	adantl 481	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌)))
18		opelxp 5724	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ↔ (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴))
19	18	anbi1i 624	. . . . . 6 ⊢ ((⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌))
20		br2ndeqg 8035	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (⟨𝑋, 𝑌⟩2nd 𝑌 ↔ 𝑌 = 𝑌))
21	20	anbi2d 630	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌)))
22	19, 21	bitrid 283	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌)))
23	17, 22	bitrd 279	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌)))
24	15, 23	bitrd 279	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌)))
25	11, 24	bitrd 279	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴) ∧ 𝑌 = 𝑌)))
26	4, 25	mpbird 257	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  Vcvv 3477  {csn 4630  ⟨cop 4636   class class class wbr 5147   × cxp 5686  ^◡ccnv 5687   ↾ cres 5690   Fn wfn 6557  –1-1-onto→wf1o 6561  ‘cfv 6562  2^nd c2nd 8011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-1st 8012  df-2nd 8013

This theorem is referenced by: (None)