Theorem 1stconst 8032

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 → (1st ↾ (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 4735	. . 3 ⊢ (𝐵 ∈ 𝑉 → {𝐵} ≠ ∅)
2		fo1stres 7947	. . 3 ⊢ ({𝐵} ≠ ∅ → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴)
3	1, 2	syl 17	. 2 ⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴)
4		moeq 3665	. . . . . 6 ⊢ ∃*𝑥 𝑥 = ⟨𝑦, 𝐵⟩
5	4	moani 2551	. . . . 5 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)
6		vex 3449	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	6	brresi 5946	. . . . . . 7 ⊢ (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦))
8		fo1st 7941	. . . . . . . . . . 11 ⊢ 1st :V–onto→V
9		fofn 6758	. . . . . . . . . . 11 ⊢ (1st :V–onto→V → 1st Fn V)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ 1st Fn V
11		vex 3449	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		fnbrfvb 6895	. . . . . . . . . 10 ⊢ ((1st Fn V ∧ 𝑥 ∈ V) → ((1st ‘𝑥) = 𝑦 ↔ 𝑥1st 𝑦))
13	10, 11, 12	mp2an 690	. . . . . . . . 9 ⊢ ((1st ‘𝑥) = 𝑦 ↔ 𝑥1st 𝑦)
14	13	anbi2i 623	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st ‘𝑥) = 𝑦) ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦))
15		elxp7 7956	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})))
16		eleq1 2825	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑥) = 𝑦 → ((1st ‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17	16	biimpac 479	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑥) ∈ 𝐴 ∧ (1st ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴)
18	17	adantlr 713	. . . . . . . . . . . . 13 ⊢ ((((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵}) ∧ (1st ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴)
19	18	adantll 712	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})) ∧ (1st ‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴)
20		elsni 4603	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑥) ∈ {𝐵} → (2nd ‘𝑥) = 𝐵)
21		eqopi 7957	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) = 𝑦 ∧ (2nd ‘𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
22	21	anass1rs 653	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (V × V) ∧ (2nd ‘𝑥) = 𝐵) ∧ (1st ‘𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
23	20, 22	sylanl2 679	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (V × V) ∧ (2nd ‘𝑥) ∈ {𝐵}) ∧ (1st ‘𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
24	23	adantlrl 718	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})) ∧ (1st ‘𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
25	19, 24	jca 512	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ {𝐵})) ∧ (1st ‘𝑥) = 𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
26	15, 25	sylanb 581	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st ‘𝑥) = 𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩))
27	26	adantl 482	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st ‘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 4620	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
31	30	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
32	29, 31	opelxpd 5671	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
33	28, 32	eqeltrd 2838	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
34	28	fveq2d 6846	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
35		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ 𝑉)
36		op1stg 7933	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
37	29, 35, 36	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
38	34, 37	eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘𝑥) = 𝑦)
39	33, 38	jca 512	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)) → (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st ‘𝑥) = 𝑦))
40	27, 39	impbida 799	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st ‘𝑥) = 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
41	14, 40	bitr3id 284	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
42	7, 41	bitrid 282	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
43	42	mobidv 2547	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝐵⟩)))
44	5, 43	mpbiri 257	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
45	44	alrimiv 1930	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
46		funcnv2 6569	. . 3 ⊢ (Fun ◡(1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
47	45, 46	sylibr 233	. 2 ⊢ (𝐵 ∈ 𝑉 → Fun ◡(1st ↾ (𝐴 × {𝐵})))
48		dff1o3 6790	. 2 ⊢ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴 ↔ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto→𝐴 ∧ Fun ◡(1st ↾ (𝐴 × {𝐵}))))
49	3, 47, 48	sylanbrc 583	1 ⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∃*wmo 2536   ≠ wne 2943  Vcvv 3445  ∅c0 4282  {csn 4586  ⟨cop 4592   class class class wbr 5105   × cxp 5631  ^◡ccnv 5632   ↾ cres 5635  Fun wfun 6490   Fn wfn 6491  –onto→wfo 6494  –1-1-onto→wf1o 6495  ‘cfv 6496  1^st c1st 7919  2^nd c2nd 7920

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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-1st 7921  df-2nd 7922

This theorem is referenced by:  curry2  8039  domss2  9080  fv1stcnv  34351