fv1stcnv - Mathbox for Scott Fenton

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Theorem fv1stcnv 35984

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 4618	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ {𝑌})
2	1	anim2i 618	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))
3		eqid 2737	. . . 4 ⊢ 𝑋 = 𝑋
4	2, 3	jctir 520	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋))
5		opex 5413	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
6		brcnvg 5829	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1^st ↾ (𝐴 × {𝑌}))𝑋))
7	5, 6	mpan2 692	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1^st ↾ (𝐴 × {𝑌}))𝑋))
8		brres 5946	. . . . . 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 5661	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))
12	11	anbi1i 625	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋))
13		br1steqg 7958	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (⟨𝑋, 𝑌⟩1^st 𝑋 ↔ 𝑋 = 𝑋))
14	13	anbi2d 631	. . . . 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 8045	. . . 4 ⊢ (𝑌 ∈ 𝑉 → (1^st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto→𝐴)
19		f1ocnv 6787	. . . 4 ⊢ ((1^st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto→𝐴 → ^◡(1^st ↾ (𝐴 × {𝑌})):𝐴–1-1-onto→(𝐴 × {𝑌}))
20		f1ofn 6776	. . . 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 6885	. . 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 687	. 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 1542 ∈ wcel 2114 Vcvv 3441 {csn 4581 ⟨cop 4587 class class class wbr 5099 × cxp 5623 ^◡ccnv 5624 ↾ cres 5627 Fn wfn 6488 –1-1-onto→wf1o 6492 ‘cfv 6493 1^st c1st 7934

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5242 ax-nul 5252 ax-pr 5378 ax-un 7683

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-1st 7936 df-2nd 7937

This theorem is referenced by: (None)

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