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Theorem 1stconst 8082

Description: The mapping of a restriction of the 1^st function to a constant function. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

**Assertion**
Ref	Expression
1stconst	⊢ (𝐵 ∈ 𝑉 → (1^st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴)

Proof of Theorem 1stconst Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snnzg 4777	. . 3 ⊢ (𝐵 ∈ 𝑉 → {𝐵} ≠ ∅)
2		fo1stres 7997	. . 3 ⊢ ({𝐵} ≠ ∅ → (1^st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴)
3	1, 2	syl 17	. 2 ⊢ (𝐵 ∈ 𝑉 → (1^st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴)
4		moeq 3702	. . . . . 6 ⊢ ∃*𝑥 𝑥 = ⟨𝑦, 𝐵⟩
5	4	moani 2547	. . . . 5 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)
6		vex 3478	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	6	brresi 5988	. . . . . . 7 ⊢ (𝑥(1^st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1^st 𝑦))
8		fo1st 7991	. . . . . . . . . . 11 ⊢ 1^st :V–onto→V
9		fofn 6804	. . . . . . . . . . 11 ⊢ (1^st :V–onto→V → 1^st Fn V)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ 1^st Fn V
11		vex 3478	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		fnbrfvb 6941	. . . . . . . . . 10 ⊢ ((1^st Fn V ∧ 𝑥 ∈ V) → ((1^st ‘𝑥) = 𝑦 ↔ 𝑥1^st 𝑦))
13	10, 11, 12	mp2an 690	. . . . . . . . 9 ⊢ ((1^st ‘𝑥) = 𝑦 ↔ 𝑥1^st 𝑦)
14	13	anbi2i 623	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1^st ‘𝑥) = 𝑦) ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1^st 𝑦))
15		elxp7 8006	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1^st ‘𝑥) ∈ 𝐴 ∧ (2^nd ‘𝑥) ∈ {𝐵})))
16		eleq1 2821	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘𝑥) = 𝑦 → ((1^st ‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17	16	biimpac 479	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘𝑥) ∈ 𝐴 ∧ (1^st ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴)
18	17	adantlr 713	. . . . . . . . . . . . 13 ⊢ ((((1^st ‘𝑥) ∈ 𝐴 ∧ (2^nd ‘𝑥) ∈ {𝐵}) ∧ (1^st ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴)
19	18	adantll 712	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (V × V) ∧ ((1^st ‘𝑥) ∈ 𝐴 ∧ (2^nd ‘𝑥) ∈ {𝐵})) ∧ (1^st ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴)
20		elsni 4644	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘𝑥) ∈ {𝐵} → (2^nd ‘𝑥) = 𝐵)
21		eqopi 8007	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (V × V) ∧ ((1^st ‘𝑥) = 𝑦 ∧ (2^nd ‘𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
22	21	anass1rs 653	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (V × V) ∧ (2^nd ‘𝑥) = 𝐵) ∧ (1^st ‘𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
23	20, 22	sylanl2 679	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (V × V) ∧ (2^nd ‘𝑥) ∈ {𝐵}) ∧ (1^st ‘𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
24	23	adantlrl 718	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (V × V) ∧ ((1^st ‘𝑥) ∈ 𝐴 ∧ (2^nd ‘𝑥) ∈ {𝐵})) ∧ (1^st ‘𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
25	19, 24	jca 512	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (V × V) ∧ ((1^st ‘𝑥) ∈ 𝐴 ∧ (2^nd ‘𝑥) ∈ {𝐵})) ∧ (1^st ‘𝑥) = 𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
26	15, 25	sylanb 581	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1^st ‘𝑥) = 𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
27	26	adantl 482	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1^st ‘𝑥) = 𝑦)) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
28		simprr 771	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
29		simprl 769	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦 ∈ 𝐴)
30		snidg 4661	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
31	30	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
32	29, 31	opelxpd 5713	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
33	28, 32	eqeltrd 2833	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
34	28	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1^st ‘𝑥) = (1^st ‘⟨𝑦, 𝐵⟩))
35		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ 𝑉)
36		op1stg 7983	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (1^st ‘⟨𝑦, 𝐵⟩) = 𝑦)
37	29, 35, 36	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1^st ‘⟨𝑦, 𝐵⟩) = 𝑦)
38	34, 37	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1^st ‘𝑥) = 𝑦)
39	33, 38	jca 512	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1^st ‘𝑥) = 𝑦))
40	27, 39	impbida 799	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1^st ‘𝑥) = 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
41	14, 40	bitr3id 284	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1^st 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
42	7, 41	bitrid 282	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝑥(1^st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
43	42	mobidv 2543	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∃𝑥 𝑥(1^st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
44	5, 43	mpbiri 257	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃*𝑥 𝑥(1^st ↾ (𝐴 × {𝐵}))𝑦)
45	44	alrimiv 1930	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∀𝑦∃*𝑥 𝑥(1^st ↾ (𝐴 × {𝐵}))𝑦)
46		funcnv2 6613	. . 3 ⊢ (Fun ^◡(1^st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1^st ↾ (𝐴 × {𝐵}))𝑦)
47	45, 46	sylibr 233	. 2 ⊢ (𝐵 ∈ 𝑉 → Fun ^◡(1^st ↾ (𝐴 × {𝐵})))
48		dff1o3 6836	. 2 ⊢ ((1^st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴 ↔ ((1^st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴 ∧ Fun ^◡(1^st ↾ (𝐴 × {𝐵}))))
49	3, 47, 48	sylanbrc 583	1 ⊢ (𝐵 ∈ 𝑉 → (1^st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∃*wmo 2532 ≠ wne 2940 Vcvv 3474 ∅c0 4321 {csn 4627 ⟨cop 4633 class class class wbr 5147 × cxp 5673 ^◡ccnv 5674 ↾ cres 5677 Fun wfun 6534 Fn wfn 6535 –onto→wfo 6538 –1-1-onto→wf1o 6539 ‘cfv 6540 1^st c1st 7969 2^nd c2nd 7970

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-1st 7971 df-2nd 7972

This theorem is referenced by: curry2 8089 domss2 9132 fv1stcnv 34736

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