Theorem fo1stresm 5730

Description: Onto mapping of a restriction of the 1^st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)

Assertion

Ref	Expression
fo1stresm	⊢ (∃y y ∈ B → (1st ↾ (A × B)):(A × B)–onto→A)

Distinct variable group:   y,B

Allowed substitution hint:   A(y)

Proof of Theorem fo1stresm

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2097	. . 3 ⊢ (v = y → (v ∈ B ↔ y ∈ B))
2	1	cbvexv 1792	. 2 ⊢ (∃v v ∈ B ↔ ∃y y ∈ B)
3		opelxp 4317	. . . . . . . . . 10 ⊢ (⟨u, v⟩ ∈ (A × B) ↔ (u ∈ A ∧ v ∈ B))
4		fvres 5141	. . . . . . . . . . . 12 ⊢ (⟨u, v⟩ ∈ (A × B) → ((1st ↾ (A × B))‘⟨u, v⟩) = (1st ‘⟨u, v⟩))
5		vex 2554	. . . . . . . . . . . . 13 ⊢ u ∈ V
6		vex 2554	. . . . . . . . . . . . 13 ⊢ v ∈ V
7	5, 6	op1st 5715	. . . . . . . . . . . 12 ⊢ (1st ‘⟨u, v⟩) = u
8	4, 7	syl6req 2086	. . . . . . . . . . 11 ⊢ (⟨u, v⟩ ∈ (A × B) → u = ((1st ↾ (A × B))‘⟨u, v⟩))
9		f1stres 5728	. . . . . . . . . . . . 13 ⊢ (1st ↾ (A × B)):(A × B)⟶A
10		ffn 4989	. . . . . . . . . . . . 13 ⊢ ((1st ↾ (A × B)):(A × B)⟶A → (1st ↾ (A × B)) Fn (A × B))
11	9, 10	ax-mp 7	. . . . . . . . . . . 12 ⊢ (1st ↾ (A × B)) Fn (A × B)
12		fnfvelrn 5242	. . . . . . . . . . . 12 ⊢ (((1st ↾ (A × B)) Fn (A × B) ∧ ⟨u, v⟩ ∈ (A × B)) → ((1st ↾ (A × B))‘⟨u, v⟩) ∈ ran (1st ↾ (A × B)))
13	11, 12	mpan 400	. . . . . . . . . . 11 ⊢ (⟨u, v⟩ ∈ (A × B) → ((1st ↾ (A × B))‘⟨u, v⟩) ∈ ran (1st ↾ (A × B)))
14	8, 13	eqeltrd 2111	. . . . . . . . . 10 ⊢ (⟨u, v⟩ ∈ (A × B) → u ∈ ran (1st ↾ (A × B)))
15	3, 14	sylbir 125	. . . . . . . . 9 ⊢ ((u ∈ A ∧ v ∈ B) → u ∈ ran (1st ↾ (A × B)))
16	15	expcom 109	. . . . . . . 8 ⊢ (v ∈ B → (u ∈ A → u ∈ ran (1st ↾ (A × B))))
17	16	exlimiv 1486	. . . . . . 7 ⊢ (∃v v ∈ B → (u ∈ A → u ∈ ran (1st ↾ (A × B))))
18	17	ssrdv 2945	. . . . . 6 ⊢ (∃v v ∈ B → A ⊆ ran (1st ↾ (A × B)))
19		frn 4995	. . . . . . 7 ⊢ ((1st ↾ (A × B)):(A × B)⟶A → ran (1st ↾ (A × B)) ⊆ A)
20	9, 19	ax-mp 7	. . . . . 6 ⊢ ran (1st ↾ (A × B)) ⊆ A
21	18, 20	jctil 295	. . . . 5 ⊢ (∃v v ∈ B → (ran (1st ↾ (A × B)) ⊆ A ∧ A ⊆ ran (1st ↾ (A × B))))
22		eqss 2954	. . . . 5 ⊢ (ran (1st ↾ (A × B)) = A ↔ (ran (1st ↾ (A × B)) ⊆ A ∧ A ⊆ ran (1st ↾ (A × B))))
23	21, 22	sylibr 137	. . . 4 ⊢ (∃v v ∈ B → ran (1st ↾ (A × B)) = A)
24	23, 9	jctil 295	. . 3 ⊢ (∃v v ∈ B → ((1st ↾ (A × B)):(A × B)⟶A ∧ ran (1st ↾ (A × B)) = A))
25		dffo2 5053	. . 3 ⊢ ((1st ↾ (A × B)):(A × B)–onto→A ↔ ((1st ↾ (A × B)):(A × B)⟶A ∧ ran (1st ↾ (A × B)) = A))
26	24, 25	sylibr 137	. 2 ⊢ (∃v v ∈ B → (1st ↾ (A × B)):(A × B)–onto→A)
27	2, 26	sylbir 125	1 ⊢ (∃y y ∈ B → (1st ↾ (A × B)):(A × B)–onto→A)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390   ⊆ wss 2911  ⟨cop 3370   × cxp 4286  ran crn 4289   ↾ cres 4290   Fn wfn 4840  ⟶wf 4841  –onto→wfo 4843  ‘cfv 4845  1^st c1st 5707

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709

This theorem is referenced by:  1stconst  5784