1stpreimas - Mathbox for Thierry Arnoux

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Theorem 1stpreimas 32797

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 7976	. . . . . . . . 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 729	. . . . . . 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 6848	. . . . . . . . . . . 12 ⊢ (1^st ‘𝑧) ∈ V
5	4	elsn 4583	. . . . . . . . . . 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 729	. . . . . . . . 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 4823	. . . . . . 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 2772	. . . . . 6 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 = ⟨𝑋, (2^nd ‘𝑧)⟩)
11		simplr 769	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑋 ∈ 𝑉)
12		simprrr 782	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋}))
13		elimasng 6049	. . . . . . . 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 838	. . . . . 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 2837	. . . . 5 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1^st ‘𝑧) ∈ {𝑋} ∧ (2^nd ‘𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 ∈ 𝐴)
17		fvres 6854	. . . . . . 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 2772	. . . . 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 5632	. . . . . . . . 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 3922	. . . . . 6 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (V × V))
25	24	adantrr 718	. . . . 5 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑧 ∈ (V × V))
26	17	ad2antrl 729	. . . . . . . 8 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ((1^st ↾ 𝐴)‘𝑧) = (1^st ‘𝑧))
27		simprr 773	. . . . . . . 8 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ((1^st ↾ 𝐴)‘𝑧) = 𝑋)
28	26, 27	eqtr3d 2774	. . . . . . 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 2838	. . . . . . . . 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 4823	. . . . . . . . . 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 2822	. . . . . . . . 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 7986	. . . . . . . . . . . 12 ⊢ ((Rel 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
35	34	ad2ant2r 748	. . . . . . . . . . 11 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
36	28	opeq1d 4823	. . . . . . . . . . 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 2772	. . . . . . . . . 10 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨𝑋, (2^nd ‘𝑧)⟩)
38		simprl 771	. . . . . . . . . 10 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → 𝑧 ∈ 𝐴)
39	37, 38	eqeltrrd 2838	. . . . . . . . 9 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ⟨𝑋, (2^nd ‘𝑧)⟩ ∈ 𝐴)
40	30, 33, 39	rspcedvd 3567	. . . . . . . 8 ⊢ (((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) ∧ (𝑧 ∈ 𝐴 ∧ ((1^st ↾ 𝐴)‘𝑧) = 𝑋)) → ∃𝑥 ∈ {𝑋}⟨𝑥, (2^nd ‘𝑧)⟩ ∈ 𝐴)
41		df-rex 3063	. . . . . . . 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 6848	. . . . . . . 8 ⊢ (2^nd ‘𝑧) ∈ V
44	43	elima3 6027	. . . . . . 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 801	. . 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 7971	. . . 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 7956	. . . . . . 7 ⊢ 1^st :V–onto→V
52		fofn 6749	. . . . . . 7 ⊢ (1^st :V–onto→V → 1^st Fn V)
53	51, 52	ax-mp 5	. . . . . 6 ⊢ 1^st Fn V
54		ssv 3947	. . . . . 6 ⊢ 𝐴 ⊆ V
55		fnssres 6616	. . . . . 6 ⊢ ((1^st Fn V ∧ 𝐴 ⊆ V) → (1^st ↾ 𝐴) Fn 𝐴)
56	53, 54, 55	mp2an 693	. . . . 5 ⊢ (1^st ↾ 𝐴) Fn 𝐴
57		fniniseg 7007	. . . . 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 2735	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 1542 ∃wex 1781 ∈ wcel 2114 ∃wrex 3062 Vcvv 3430 ⊆ wss 3890 {csn 4568 ⟨cop 4574 × cxp 5623 ^◡ccnv 5624 ↾ cres 5627 “ cima 5628 Rel wrel 5630 Fn wfn 6488 –onto→wfo 6491 ‘cfv 6493 1^st c1st 7934 2^nd c2nd 7935

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 5232 ax-nul 5242 ax-pr 5371 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 3063 df-rab 3391 df-v 3432 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 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-fo 6499 df-fv 6501 df-1st 7936 df-2nd 7937

This theorem is referenced by: gsummpt2d 33128

W3C validator