fv1stcnv - Mathbox for Scott Fenton

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Theorem fv1stcnv 35053

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) → (^◡(1^st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)

**Proof of Theorem fv1stcnv**
Step	Hyp	Ref	Expression
1		snidg 4662	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ {𝑌})
2	1	anim2i 616	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))
3		eqid 2731	. . . 4 ⊢ 𝑋 = 𝑋
4	2, 3	jctir 520	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋))
5		opex 5464	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
6		brcnvg 5879	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1^st ↾ (𝐴 × {𝑌}))𝑋))
7	5, 6	mpan2 688	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1^st ↾ (𝐴 × {𝑌}))𝑋))
8		brres 5988	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (⟨𝑋, 𝑌⟩(1^st ↾ (𝐴 × {𝑌}))𝑋 ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋)))
9	7, 8	bitrd 279	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋)))
10	9	adantr 480	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋)))
11		opelxp 5712	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))
12	11	anbi1i 623	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋))
13		br1steqg 8000	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (⟨𝑋, 𝑌⟩1^st 𝑋 ↔ 𝑋 = 𝑋))
14	13	anbi2d 628	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
15	12, 14	bitrid 283	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
16	10, 15	bitrd 279	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
17	4, 16	mpbird 257	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → 𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩)
18		1stconst 8089	. . . 4 ⊢ (𝑌 ∈ 𝑉 → (1^st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto→𝐴)
19		f1ocnv 6845	. . . 4 ⊢ ((1^st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto→𝐴 → ^◡(1^st ↾ (𝐴 × {𝑌})):𝐴–1-1-onto→(𝐴 × {𝑌}))
20		f1ofn 6834	. . . 4 ⊢ (^◡(1^st ↾ (𝐴 × {𝑌})):𝐴–1-1-onto→(𝐴 × {𝑌}) → ^◡(1^st ↾ (𝐴 × {𝑌})) Fn 𝐴)
21	18, 19, 20	3syl 18	. . 3 ⊢ (𝑌 ∈ 𝑉 → ^◡(1^st ↾ (𝐴 × {𝑌})) Fn 𝐴)
22		simpl 482	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ 𝐴)
23		fnbrfvb 6944	. . 3 ⊢ ((^◡(1^st ↾ (𝐴 × {𝑌})) Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((^◡(1^st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
24	21, 22, 23	syl2an2 683	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((^◡(1^st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
25	17, 24	mpbird 257	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (^◡(1^st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 {csn 4628 ⟨cop 4634 class class class wbr 5148 × cxp 5674 ^◡ccnv 5675 ↾ cres 5678 Fn wfn 6538 –1-1-onto→wf1o 6542 ‘cfv 6543 1^st c1st 7976

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7728

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-1st 7978 df-2nd 7979

This theorem is referenced by: (None)

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