fv1stcnv - Mathbox for Scott Fenton

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Theorem fv1stcnv 35771

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 4627	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ {𝑌})
2	1	anim2i 617	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))
3		eqid 2730	. . . 4 ⊢ 𝑋 = 𝑋
4	2, 3	jctir 520	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋))
5		opex 5427	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
6		brcnvg 5846	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1^st ↾ (𝐴 × {𝑌}))𝑋))
7	5, 6	mpan2 691	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1^st ↾ (𝐴 × {𝑌}))𝑋))
8		brres 5960	. . . . . 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 5677	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))
12	11	anbi1i 624	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋))
13		br1steqg 7993	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (⟨𝑋, 𝑌⟩1^st 𝑋 ↔ 𝑋 = 𝑋))
14	13	anbi2d 630	. . . . 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 8082	. . . 4 ⊢ (𝑌 ∈ 𝑉 → (1^st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto→𝐴)
19		f1ocnv 6815	. . . 4 ⊢ ((1^st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto→𝐴 → ^◡(1^st ↾ (𝐴 × {𝑌})):𝐴–1-1-onto→(𝐴 × {𝑌}))
20		f1ofn 6804	. . . 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 6914	. . 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 686	. 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3450 {csn 4592 ⟨cop 4598 class class class wbr 5110 × cxp 5639 ^◡ccnv 5640 ↾ cres 5643 Fn wfn 6509 –1-1-onto→wf1o 6513 ‘cfv 6514 1^st c1st 7969

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-1st 7971 df-2nd 7972

This theorem is referenced by: (None)

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