Theorem fo2ndresm 6217

Description: Onto mapping of a restriction of the 2^nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)

Assertion

Ref	Expression
fo2ndresm	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem fo2ndresm

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

Step	Hyp	Ref	Expression
1		eleq1 2256	. . 3 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
2	1	cbvexv 1930	. 2 ⊢ (∃𝑢 𝑢 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴)
3		opelxp 4690	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵))
4		fvres 5579	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (2nd ‘⟨𝑢, 𝑣⟩))
5		vex 2763	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
6		vex 2763	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
7	5, 6	op2nd 6202	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨𝑢, 𝑣⟩) = 𝑣
8	4, 7	eqtr2di 2243	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 = ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9		f2ndres 6215	. . . . . . . . . . . . 13 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
10		ffn 5404	. . . . . . . . . . . . 13 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
11	9, 10	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
12		fnfvelrn 5691	. . . . . . . . . . . 12 ⊢ (((2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
13	11, 12	mpan 424	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
14	8, 13	eqeltrd 2270	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
15	3, 14	sylbir 135	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
16	15	ex 115	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐴 → (𝑣 ∈ 𝐵 → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
17	16	exlimiv 1609	. . . . . . 7 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → (𝑣 ∈ 𝐵 → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
18	17	ssrdv 3186	. . . . . 6 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵)))
19		frn 5413	. . . . . . 7 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵)
20	9, 19	ax-mp 5	. . . . . 6 ⊢ ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵
21	18, 20	jctil 312	. . . . 5 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
22		eqss 3195	. . . . 5 ⊢ (ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
23	21, 22	sylibr 134	. . . 4 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵)
24	23, 9	jctil 312	. . 3 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
25		dffo2 5481	. . 3 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
26	24, 25	sylibr 134	. 2 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)
27	2, 26	sylbir 135	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wex 1503   ∈ wcel 2164   ⊆ wss 3154  ⟨cop 3622   × cxp 4658  ran crn 4661   ↾ cres 4662   Fn wfn 5250  ⟶wf 5251  –onto→wfo 5253  ‘cfv 5255  2^nd c2nd 6194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-2nd 6196

This theorem is referenced by:  2ndconst  6277