Theorem fv2ndcnv 30926

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 4153	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
2	1	anim1i 590	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴))
3		eqid 2610	. . 3 ⊢ 𝑌 = 𝑌
4	2, 3	jctil 558	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌 = 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴)))
5		2ndconst 7153	. . . . . 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 6062	. . . . 5 ⊢ ((2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto→𝐴 → ◡(2nd ↾ ({𝑋} × 𝐴)):𝐴–1-1-onto→({𝑋} × 𝐴))
8		f1ofn 6051	. . . . 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 6146	. . . 4 ⊢ ((◡(2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴 ∧ 𝑌 ∈ 𝐴) → ((◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
11	9, 10	sylancom 698	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
12		opex 4859	. . . . . 6 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
13		brcnvg 5225	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
14	12, 13	mpan2 703	. . . . 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		brresg 5325	. . . . . 6 ⊢ (𝑌 ∈ 𝐴 → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩2nd 𝑌 ∧ ⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴))))
17	16	adantl 481	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩2nd 𝑌 ∧ ⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴))))
18		opelxp 5070	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ↔ (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴))
19	18	anbi2i 726	. . . . . 6 ⊢ ((⟨𝑋, 𝑌⟩2nd 𝑌 ∧ ⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴)) ↔ (⟨𝑋, 𝑌⟩2nd 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴)))
20		br2ndeqg 30920	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (⟨𝑋, 𝑌⟩2nd 𝑌 ↔ 𝑌 = 𝑌))
21	20	3anidm23 1377	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (⟨𝑋, 𝑌⟩2nd 𝑌 ↔ 𝑌 = 𝑌))
22	21	anbi1d 737	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((⟨𝑋, 𝑌⟩2nd 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴)) ↔ (𝑌 = 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴))))
23	19, 22	syl5bb 271	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((⟨𝑋, 𝑌⟩2nd 𝑌 ∧ ⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴)) ↔ (𝑌 = 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴))))
24	17, 23	bitrd 267	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (𝑌 = 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴))))
25	15, 24	bitrd 267	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌◡(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ (𝑌 = 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴))))
26	11, 25	bitrd 267	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ (𝑌 = 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝐴))))
27	4, 26	mpbird 246	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ^◡ccnv 5037   ↾ cres 5040   Fn wfn 5799  –1-1-onto→wf1o 5803  ‘cfv 5804  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: (None)