1stpreimas - Mathbox for Thierry Arnoux

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Theorem 1stpreimas 33299

Description: The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.)

**Assertion**
Ref	Expression
1stpreimas	⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (^◡(1^st ↾ 𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))

Proof of Theorem 1stpreimas Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1st2ndb 8041	. . . . . . . . 9 ⊢ (𝑧 ∈ (V × V) ↔ 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
2	1	biimpi 219	. . . . . . . 8 ⊢ (𝑧 ∈ (V × V) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
3	2	ad2antrl 741	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
4		fvex 6898	. . . . . . . . . . . 12 ⊢ (1^st ‘𝑧) ∈ V
5	4	elsn 4599	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑧) ∈ {𝑋} ↔ (1^st ‘𝑧) = 𝑋)
6	5	biimpi 219	. . . . . . . . . 10 ⊢ ((1^st ‘𝑧) ∈ {𝑋} → (1^st ‘𝑧) = 𝑋)
7	6	ad2antrl 741	. . . . . . . . 9 ⊢ ((𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}))) → (1^st ‘𝑧) = 𝑋)
8	7	adantl 487	. . . . . . . 8 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → (1^st ‘𝑧) = 𝑋)
9	8	opeq1d 4839	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ = ⟨𝑋, (2^nd ‘𝑧)⟩)
10	3, 9	eqtrd 2796	. . . . . 6 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 = ⟨𝑋, (2^nd ‘𝑧)⟩)
11		simplr 781	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑋 ∈ 𝑉)
12		simprrr 794	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}))
13		elimasng 6087	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})) → ((2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}) ↔ ⟨𝑋, (2^nd ‘𝑧)⟩ ∈ 𝐴))
14	13	biimpa 482	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})) ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})) → ⟨𝑋, (2^nd ‘𝑧)⟩ ∈ 𝐴)
15	11, 12, 12, 14	syl21anc 851	. . . . . 6 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → ⟨𝑋, (2^nd ‘𝑧)⟩ ∈ 𝐴)
16	10, 15	eqeltrd 2861	. . . . 5 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 ∈ 𝐴)
17		fvres 6904	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((1^st ↾ 𝐴)‘𝑧) = (1^st ‘𝑧))
18	16, 17	syl 18	. . . . . 6 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → ((1^st ↾ 𝐴)‘𝑧) = (1^st ‘𝑧))
19	18, 8	eqtrd 2796	. . . . 5 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → ((1^st ↾ 𝐴)‘𝑧) = 𝑋)
20	16, 19	jca 521	. . . 4 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋))
21		df-rel 5658	. . . . . . . 8 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
22	21	birani 509	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → 𝐴 ⊆ (V × V))
23	22	sselda 3931	. . . . . 6 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (V × V))
24	23	adantrr 730	. . . . 5 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑧 ∈ (V × V))
25	17	ad2antrl 741	. . . . . . . 8 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ((1^st ↾ 𝐴)‘𝑧) = (1^st ‘𝑧))
26		simprr 785	. . . . . . . 8 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ((1^st ↾ 𝐴)‘𝑧) = 𝑋)
27	25, 26	eqtr3d 2798	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → (1^st ‘𝑧) = 𝑋)
28	27, 5	sylibr 237	. . . . . 6 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → (1^st ‘𝑧) ∈ {𝑋})
29	27, 28	eqeltrrd 2862	. . . . . . . . 9 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑋 ∈ {𝑋})
30		simpr 490	. . . . . . . . . . 11 ⊢ ((((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
31	30	opeq1d 4839	. . . . . . . . . 10 ⊢ ((((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → ⟨𝑥, (2^nd ‘𝑧)⟩ = ⟨𝑋, (2^nd ‘𝑧)⟩)
32	31	eleq1d 2846	. . . . . . . . 9 ⊢ ((((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → (⟨𝑥, (2^nd ‘𝑧)⟩ ∈ 𝐴 ↔ ⟨𝑋, (2^nd ‘𝑧)⟩ ∈ 𝐴))
33		1st2nd 8050	. . . . . . . . . . . 12 ⊢ ((Rel 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
34	33	ad2ant2r 760	. . . . . . . . . . 11 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
35	27	opeq1d 4839	. . . . . . . . . . 11 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ = ⟨𝑋, (2^nd ‘𝑧)⟩)
36	34, 35	eqtrd 2796	. . . . . . . . . 10 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨𝑋, (2^nd ‘𝑧)⟩)
37		simprl 783	. . . . . . . . . 10 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑧 ∈ 𝐴)
38	36, 37	eqeltrrd 2862	. . . . . . . . 9 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ⟨𝑋, (2^nd ‘𝑧)⟩ ∈ 𝐴)
39	29, 32, 38	rspcedvd 3579	. . . . . . . 8 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ∃𝑥 ∈ {𝑋}⟨𝑥, (2^nd ‘𝑧)⟩ ∈ 𝐴)
40		df-rex 3088	. . . . . . . 8 ⊢ (∃𝑥 ∈ {𝑋}⟨𝑥, (2^nd ‘𝑧)⟩ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2^nd ‘𝑧)⟩ ∈ 𝐴))
41	39, 40	sylib 221	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2^nd ‘𝑧)⟩ ∈ 𝐴))
42		fvex 6898	. . . . . . . 8 ⊢ (2^nd ‘𝑧) ∈ V
43	42	elima3 6063	. . . . . . 7 ⊢ ((2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}) ↔ ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2^nd ‘𝑧)⟩ ∈ 𝐴))
44	41, 43	sylibr 237	. . . . . 6 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}))
45	28, 44	jca 521	. . . . 5 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))
46	24, 45	jca 521	. . . 4 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}))))
47	20, 46	impbida 813	. . 3 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → ((𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}))) ↔ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)))
48		elxp7 8036	. . . 4 ⊢ (𝑧 ∈ ({𝑋} × (𝐴 “ {𝑋})) ↔ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}))))
49	48	a1i 11	. . 3 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (𝑧 ∈ ({𝑋} × (𝐴 “ {𝑋})) ↔ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))))
50		fo1st 8021	. . . . . . 7 ⊢ 1^st :V–onto→V
51		fofn 6798	. . . . . . 7 ⊢ (1^st :V–onto→V → 1^st Fn V)
52	50, 51	ax-mp 5	. . . . . 6 ⊢ 1^st Fn V
53		ssv 3955	. . . . . 6 ⊢ 𝐴 ⊆ V
54		fnssres 6662	. . . . . 6 ⊢ ((1^st Fn V ∧ 𝐴 ⊆ V) → (1^st ↾ 𝐴) Fn 𝐴)
55	52, 53, 54	mp2an 705	. . . . 5 ⊢ (1^st ↾ 𝐴) Fn 𝐴
56		fniniseg 7059	. . . . 5 ⊢ ((1^st ↾ 𝐴) Fn 𝐴 → (𝑧 ∈ (^◡(1^st ↾ 𝐴) “ {𝑋}) ↔ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)))
57	55, 56	ax-mp 5	. . . 4 ⊢ (𝑧 ∈ (^◡(1^st ↾ 𝐴) “ {𝑋}) ↔ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋))
58	57	a1i 11	. . 3 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (𝑧 ∈ (^◡(1^st ↾ 𝐴) “ {𝑋}) ↔ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)))
59	47, 49, 58	3bitr4rd 315	. 2 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (𝑧 ∈ (^◡(1^st ↾ 𝐴) “ {𝑋}) ↔ 𝑧 ∈ ({𝑋} × (𝐴 “ {𝑋}))))
60	59	eqrdv 2759	1 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (^◡(1^st ↾ 𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ∃wrex 3087 Vcvv 3451 ⊆ wss 3899 {csn 4584 ⟨cop 4590 × cxp 5649 ^◡ccnv 5650 ↾ cres 5653 “ cima 5654 Rel wrel 5656 Fn wfn 6533 –onto→wfo 6536 ‘cfv 6538 1^st c1st 7999 2^nd c2nd 8000

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-fo 6544 df-fv 6546 df-1st 8001 df-2nd 8002

This theorem is used by: gsummpt2d 33610

W3C validator