Theorem fo1stresm 5711

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 2082	. . 3 ⊢ (v = y → (v ∈ B ↔ y ∈ B))
2	1	cbvexv 1777	. 2 ⊢ (∃v v ∈ B ↔ ∃y y ∈ B)
3		opelxp 4301	. . . . . . . . . 10 ⊢ (⟨u, v⟩ ∈ (A × B) ↔ (u ∈ A ∧ v ∈ B))
4		fvres 5123	. . . . . . . . . . . 12 ⊢ (⟨u, v⟩ ∈ (A × B) → ((1st ↾ (A × B))‘⟨u, v⟩) = (1st ‘⟨u, v⟩))
5		vex 2538	. . . . . . . . . . . . 13 ⊢ u ∈ V
6		vex 2538	. . . . . . . . . . . . 13 ⊢ v ∈ V
7	5, 6	op1st 5696	. . . . . . . . . . . 12 ⊢ (1st ‘⟨u, v⟩) = u
8	4, 7	syl6req 2071	. . . . . . . . . . 11 ⊢ (⟨u, v⟩ ∈ (A × B) → u = ((1st ↾ (A × B))‘⟨u, v⟩))
9		f1stres 5709	. . . . . . . . . . . . 13 ⊢ (1st ↾ (A × B)):(A × B)⟶A
10		ffn 4972	. . . . . . . . . . . . 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 5224	. . . . . . . . . . . 12 ⊢ (((1st ↾ (A × B)) Fn (A × B) ∧ ⟨u, v⟩ ∈ (A × B)) → ((1st ↾ (A × B))‘⟨u, v⟩) ∈ ran (1st ↾ (A × B)))
13	11, 12	mpan 402	. . . . . . . . . . 11 ⊢ (⟨u, v⟩ ∈ (A × B) → ((1st ↾ (A × B))‘⟨u, v⟩) ∈ ran (1st ↾ (A × B)))
14	8, 13	eqeltrd 2096	. . . . . . . . . 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 1471	. . . . . . 7 ⊢ (∃v v ∈ B → (u ∈ A → u ∈ ran (1st ↾ (A × B))))
18	17	ssrdv 2928	. . . . . 6 ⊢ (∃v v ∈ B → A ⊆ ran (1st ↾ (A × B)))
19		frn 4978	. . . . . . 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 2937	. . . . 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 5035	. . 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 1228  ∃wex 1362   ∈ wcel 1374   ⊆ wss 2894  ⟨cop 3353   × cxp 4270  ran crn 4273   ↾ cres 4274   Fn wfn 4824  ⟶wf 4825  –onto→wfo 4827  ‘cfv 4829  1^st c1st 5688

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-1st 5690

This theorem is referenced by:  1stconst  5765