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Theorem 1stconst 8086

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 4773	. . 3 ⊢ (𝐵 ∈ 𝑉 → {𝐵} ≠ ∅)
2		fo1stres 8000	. . 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 3698	. . . . . 6 ⊢ ∃*𝑥 𝑥 = ⟨𝑦, 𝐵⟩
5	4	moani 2541	. . . . 5 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)
6		vex 3472	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	6	brresi 5984	. . . . . . 7 ⊢ (𝑥(1^st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1^st 𝑦))
8		fo1st 7994	. . . . . . . . . . 11 ⊢ 1^st :V–onto→V
9		fofn 6801	. . . . . . . . . . 11 ⊢ (1^st :V–onto→V → 1^st Fn V)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ 1^st Fn V
11		vex 3472	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		fnbrfvb 6938	. . . . . . . . . 10 ⊢ ((1^st Fn V ∧ 𝑥 ∈ V) → ((1^st ‘𝑥) = 𝑦 ↔ 𝑥1^st 𝑦))
13	10, 11, 12	mp2an 689	. . . . . . . . 9 ⊢ ((1^st ‘𝑥) = 𝑦 ↔ 𝑥1^st 𝑦)
14	13	anbi2i 622	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1^st ‘𝑥) = 𝑦) ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1^st 𝑦))
15		elxp7 8009	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1^st ‘𝑥) ∈ 𝐴 ∧ (2^nd ‘𝑥) ∈ {𝐵})))
16		eleq1 2815	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘𝑥) = 𝑦 → ((1^st ‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17	16	biimpac 478	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘𝑥) ∈ 𝐴 ∧ (1^st ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴)
18	17	adantlr 712	. . . . . . . . . . . . 13 ⊢ ((((1^st ‘𝑥) ∈ 𝐴 ∧ (2^nd ‘𝑥) ∈ {𝐵}) ∧ (1^st ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴)
19	18	adantll 711	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (V × V) ∧ ((1^st ‘𝑥) ∈ 𝐴 ∧ (2^nd ‘𝑥) ∈ {𝐵})) ∧ (1^st ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴)
20		elsni 4640	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘𝑥) ∈ {𝐵} → (2^nd ‘𝑥) = 𝐵)
21		eqopi 8010	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (V × V) ∧ ((1^st ‘𝑥) = 𝑦 ∧ (2^nd ‘𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
22	21	anass1rs 652	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (V × V) ∧ (2^nd ‘𝑥) = 𝐵) ∧ (1^st ‘𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
23	20, 22	sylanl2 678	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (V × V) ∧ (2^nd ‘𝑥) ∈ {𝐵}) ∧ (1^st ‘𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
24	23	adantlrl 717	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (V × V) ∧ ((1^st ‘𝑥) ∈ 𝐴 ∧ (2^nd ‘𝑥) ∈ {𝐵})) ∧ (1^st ‘𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
25	19, 24	jca 511	. . . . . . . . . . 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 580	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1^st ‘𝑥) = 𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
27	26	adantl 481	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1^st ‘𝑥) = 𝑦)) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
28		simprr 770	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
29		simprl 768	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦 ∈ 𝐴)
30		snidg 4657	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
31	30	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
32	29, 31	opelxpd 5708	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
33	28, 32	eqeltrd 2827	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
34	28	fveq2d 6889	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1^st ‘𝑥) = (1^st ‘⟨𝑦, 𝐵⟩))
35		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ 𝑉)
36		op1stg 7986	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (1^st ‘⟨𝑦, 𝐵⟩) = 𝑦)
37	29, 35, 36	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1^st ‘⟨𝑦, 𝐵⟩) = 𝑦)
38	34, 37	eqtrd 2766	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1^st ‘𝑥) = 𝑦)
39	33, 38	jca 511	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1^st ‘𝑥) = 𝑦))
40	27, 39	impbida 798	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1^st ‘𝑥) = 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
41	14, 40	bitr3id 285	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1^st 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
42	7, 41	bitrid 283	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝑥(1^st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
43	42	mobidv 2537	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∃𝑥 𝑥(1^st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
44	5, 43	mpbiri 258	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃*𝑥 𝑥(1^st ↾ (𝐴 × {𝐵}))𝑦)
45	44	alrimiv 1922	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∀𝑦∃*𝑥 𝑥(1^st ↾ (𝐴 × {𝐵}))𝑦)
46		funcnv2 6610	. . 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 6833	. 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 582	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 395 ∀wal 1531 = wceq 1533 ∈ wcel 2098 ∃*wmo 2526 ≠ wne 2934 Vcvv 3468 ∅c0 4317 {csn 4623 ⟨cop 4629 class class class wbr 5141 × cxp 5667 ^◡ccnv 5668 ↾ cres 5671 Fun wfun 6531 Fn wfn 6532 –onto→wfo 6535 –1-1-onto→wf1o 6536 ‘cfv 6537 1^st c1st 7972 2^nd c2nd 7973

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-1st 7974 df-2nd 7975

This theorem is referenced by: curry2 8093 domss2 9138 fv1stcnv 35281

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