1stpreimas - Mathbox for Thierry Arnoux

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Theorem 1stpreimas 32717

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 8070	. . . . . . . . 9 ⊢ (𝑧 ∈ (V × V) ↔ 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
2	1	biimpi 216	. . . . . . . 8 ⊢ (𝑧 ∈ (V × V) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
3	2	ad2antrl 727	. . . . . . 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 6933	. . . . . . . . . . . 12 ⊢ (1^st ‘𝑧) ∈ V
5	4	elsn 4663	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑧) ∈ {𝑋} ↔ (1^st ‘𝑧) = 𝑋)
6	5	biimpi 216	. . . . . . . . . 10 ⊢ ((1^st ‘𝑧) ∈ {𝑋} → (1^st ‘𝑧) = 𝑋)
7	6	ad2antrl 727	. . . . . . . . 9 ⊢ ((𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}))) → (1^st ‘𝑧) = 𝑋)
8	7	adantl 481	. . . . . . . 8 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → (1^st ‘𝑧) = 𝑋)
9	8	opeq1d 4903	. . . . . . 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 2780	. . . . . 6 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 = ⟨𝑋, (2^nd ‘𝑧)⟩)
11		simplr 768	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑋 ∈ 𝑉)
12		simprrr 781	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}))
13		elimasng 6118	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})) → ((2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}) ↔ ⟨𝑋, (2^nd ‘𝑧)⟩ ∈ 𝐴))
14	13	biimpa 476	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})) ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})) → ⟨𝑋, (2^nd ‘𝑧)⟩ ∈ 𝐴)
15	11, 12, 12, 14	syl21anc 837	. . . . . 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 2844	. . . . 5 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 ∈ 𝐴)
17		fvres 6939	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((1^st ↾ 𝐴)‘𝑧) = (1^st ‘𝑧))
18	16, 17	syl 17	. . . . . 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 2780	. . . . 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 511	. . . 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 5707	. . . . . . . . 9 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
22	21	biimpi 216	. . . . . . . 8 ⊢ (Rel 𝐴 → 𝐴 ⊆ (V × V))
23	22	adantr 480	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → 𝐴 ⊆ (V × V))
24	23	sselda 4008	. . . . . 6 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (V × V))
25	24	adantrr 716	. . . . 5 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑧 ∈ (V × V))
26	17	ad2antrl 727	. . . . . . . 8 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ((1^st ↾ 𝐴)‘𝑧) = (1^st ‘𝑧))
27		simprr 772	. . . . . . . 8 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ((1^st ↾ 𝐴)‘𝑧) = 𝑋)
28	26, 27	eqtr3d 2782	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → (1^st ‘𝑧) = 𝑋)
29	28, 5	sylibr 234	. . . . . 6 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → (1^st ‘𝑧) ∈ {𝑋})
30	28, 29	eqeltrrd 2845	. . . . . . . . 9 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑋 ∈ {𝑋})
31		simpr 484	. . . . . . . . . . 11 ⊢ ((((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
32	31	opeq1d 4903	. . . . . . . . . 10 ⊢ ((((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → ⟨𝑥, (2^nd ‘𝑧)⟩ = ⟨𝑋, (2^nd ‘𝑧)⟩)
33	32	eleq1d 2829	. . . . . . . . 9 ⊢ ((((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → (⟨𝑥, (2^nd ‘𝑧)⟩ ∈ 𝐴 ↔ ⟨𝑋, (2^nd ‘𝑧)⟩ ∈ 𝐴))
34		1st2nd 8080	. . . . . . . . . . . 12 ⊢ ((Rel 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
35	34	ad2ant2r 746	. . . . . . . . . . 11 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
36	28	opeq1d 4903	. . . . . . . . . . 11 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ = ⟨𝑋, (2^nd ‘𝑧)⟩)
37	35, 36	eqtrd 2780	. . . . . . . . . 10 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨𝑋, (2^nd ‘𝑧)⟩)
38		simprl 770	. . . . . . . . . 10 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑧 ∈ 𝐴)
39	37, 38	eqeltrrd 2845	. . . . . . . . 9 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ⟨𝑋, (2^nd ‘𝑧)⟩ ∈ 𝐴)
40	30, 33, 39	rspcedvd 3637	. . . . . . . 8 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ∃𝑥 ∈ {𝑋}⟨𝑥, (2^nd ‘𝑧)⟩ ∈ 𝐴)
41		df-rex 3077	. . . . . . . 8 ⊢ (∃𝑥 ∈ {𝑋}⟨𝑥, (2^nd ‘𝑧)⟩ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2^nd ‘𝑧)⟩ ∈ 𝐴))
42	40, 41	sylib 218	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2^nd ‘𝑧)⟩ ∈ 𝐴))
43		fvex 6933	. . . . . . . 8 ⊢ (2^nd ‘𝑧) ∈ V
44	43	elima3 6096	. . . . . . 7 ⊢ ((2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}) ↔ ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2^nd ‘𝑧)⟩ ∈ 𝐴))
45	42, 44	sylibr 234	. . . . . 6 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}))
46	29, 45	jca 511	. . . . 5 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))
47	25, 46	jca 511	. . . 4 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}))))
48	20, 47	impbida 800	. . 3 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → ((𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}))) ↔ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)))
49		elxp7 8065	. . . 4 ⊢ (𝑧 ∈ ({𝑋} × (𝐴 “ {𝑋})) ↔ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}))))
50	49	a1i 11	. . 3 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (𝑧 ∈ ({𝑋} × (𝐴 “ {𝑋})) ↔ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))))
51		fo1st 8050	. . . . . . 7 ⊢ 1^st :V–onto→V
52		fofn 6836	. . . . . . 7 ⊢ (1^st :V–onto→V → 1^st Fn V)
53	51, 52	ax-mp 5	. . . . . 6 ⊢ 1^st Fn V
54		ssv 4033	. . . . . 6 ⊢ 𝐴 ⊆ V
55		fnssres 6703	. . . . . 6 ⊢ ((1^st Fn V ∧ 𝐴 ⊆ V) → (1^st ↾ 𝐴) Fn 𝐴)
56	53, 54, 55	mp2an 691	. . . . 5 ⊢ (1^st ↾ 𝐴) Fn 𝐴
57		fniniseg 7093	. . . . 5 ⊢ ((1^st ↾ 𝐴) Fn 𝐴 → (𝑧 ∈ (^◡(1^st ↾ 𝐴) “ {𝑋}) ↔ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)))
58	56, 57	ax-mp 5	. . . 4 ⊢ (𝑧 ∈ (^◡(1^st ↾ 𝐴) “ {𝑋}) ↔ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋))
59	58	a1i 11	. . 3 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (𝑧 ∈ (^◡(1^st ↾ 𝐴) “ {𝑋}) ↔ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)))
60	48, 50, 59	3bitr4rd 312	. 2 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (𝑧 ∈ (^◡(1^st ↾ 𝐴) “ {𝑋}) ↔ 𝑧 ∈ ({𝑋} × (𝐴 “ {𝑋}))))
61	60	eqrdv 2738	1 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (^◡(1^st ↾ 𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1777 ∈ wcel 2108 ∃wrex 3076 Vcvv 3488 ⊆ wss 3976 {csn 4648 ⟨cop 4654 × cxp 5698 ^◡ccnv 5699 ↾ cres 5702 “ cima 5703 Rel wrel 5705 Fn wfn 6568 –onto→wfo 6571 ‘cfv 6573 1^st c1st 8028 2^nd c2nd 8029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fo 6579 df-fv 6581 df-1st 8030 df-2nd 8031

This theorem is referenced by: gsummpt2d 33032

W3C validator