Theorem fv1stcnv 31680

Description: The value of the converse of 1^st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)

Assertion

Ref	Expression
fv1stcnv	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (◡(1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)

Proof of Theorem fv1stcnv

Step	Hyp	Ref	Expression
1		snidg 4206	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ {𝑌})
2	1	anim2i 593	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))
3		eqid 2622	. . . 4 ⊢ 𝑋 = 𝑋
4	2, 3	jctil 560	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋 = 𝑋 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌})))
5		opex 4932	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
6		brcnvg 5303	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
7	5, 6	mpan2 707	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
8		brresg 5405	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋 ↔ (⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}))))
9	7, 8	bitrd 268	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}))))
10	9	adantr 481	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}))))
11		opelxp 5146	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))
12	11	anbi2i 730	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌})) ↔ (⟨𝑋, 𝑌⟩1st 𝑋 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌})))
13		br1steqg 31672	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (⟨𝑋, 𝑌⟩1st 𝑋 ↔ 𝑋 = 𝑋))
14	13	anbi1d 741	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((⟨𝑋, 𝑌⟩1st 𝑋 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌})) ↔ (𝑋 = 𝑋 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))))
15	12, 14	syl5bb 272	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌})) ↔ (𝑋 = 𝑋 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))))
16	10, 15	bitrd 268	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (𝑋 = 𝑋 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))))
17	4, 16	mpbird 247	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → 𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩)
18		1stconst 7265	. . . 4 ⊢ (𝑌 ∈ 𝑉 → (1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto→𝐴)
19		f1ocnv 6149	. . . 4 ⊢ ((1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto→𝐴 → ◡(1st ↾ (𝐴 × {𝑌})):𝐴–1-1-onto→(𝐴 × {𝑌}))
20		f1ofn 6138	. . . 4 ⊢ (◡(1st ↾ (𝐴 × {𝑌})):𝐴–1-1-onto→(𝐴 × {𝑌}) → ◡(1st ↾ (𝐴 × {𝑌})) Fn 𝐴)
21	18, 19, 20	3syl 18	. . 3 ⊢ (𝑌 ∈ 𝑉 → ◡(1st ↾ (𝐴 × {𝑌})) Fn 𝐴)
22		simpl 473	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ 𝐴)
23		fnbrfvb 6236	. . 3 ⊢ ((◡(1st ↾ (𝐴 × {𝑌})) Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((◡(1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
24	21, 22, 23	syl2an2 875	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((◡(1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
25	17, 24	mpbird 247	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (◡(1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177  ⟨cop 4183   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113   ↾ cres 5116   Fn wfn 5883  –1-1-onto→wf1o 5887  ‘cfv 5888  1^st c1st 7166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by: (None)