Theorem fv1stcnv 34351

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 4620	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ {𝑌})
2	1	anim2i 617	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))
3		eqid 2736	. . . 4 ⊢ 𝑋 = 𝑋
4	2, 3	jctir 521	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋))
5		opex 5421	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
6		brcnvg 5835	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
7	5, 6	mpan2 689	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
8		brres 5944	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋 ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋)))
9	7, 8	bitrd 278	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋)))
10	9	adantr 481	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋)))
11		opelxp 5669	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))
12	11	anbi1i 624	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋))
13		br1steqg 7943	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (⟨𝑋, 𝑌⟩1st 𝑋 ↔ 𝑋 = 𝑋))
14	13	anbi2d 629	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
15	12, 14	bitrid 282	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
16	10, 15	bitrd 278	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
17	4, 16	mpbird 256	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → 𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩)
18		1stconst 8032	. . . 4 ⊢ (𝑌 ∈ 𝑉 → (1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto→𝐴)
19		f1ocnv 6796	. . . 4 ⊢ ((1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto→𝐴 → ◡(1st ↾ (𝐴 × {𝑌})):𝐴–1-1-onto→(𝐴 × {𝑌}))
20		f1ofn 6785	. . . 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 483	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ 𝐴)
23		fnbrfvb 6895	. . 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 684	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((◡(1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋◡(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
25	17, 24	mpbird 256	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (◡(1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3445  {csn 4586  ⟨cop 4592   class class class wbr 5105   × cxp 5631  ^◡ccnv 5632   ↾ cres 5635   Fn wfn 6491  –1-1-onto→wf1o 6495  ‘cfv 6496  1^st c1st 7919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-1st 7921  df-2nd 7922

This theorem is referenced by: (None)