Theorem f2ndres 5729

Description: Mapping of a restriction of the 2^nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
f2ndres	⊢ (2nd ↾ (A × B)):(A × B)⟶B

Proof of Theorem f2ndres

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2554	. . . . . . . 8 ⊢ y ∈ V
2		vex 2554	. . . . . . . 8 ⊢ z ∈ V
3	1, 2	op2nda 4748	. . . . . . 7 ⊢ ∪ ran {⟨y, z⟩} = z
4	3	eleq1i 2100	. . . . . 6 ⊢ (∪ ran {⟨y, z⟩} ∈ B ↔ z ∈ B)
5	4	biimpri 124	. . . . 5 ⊢ (z ∈ B → ∪ ran {⟨y, z⟩} ∈ B)
6	5	adantl 262	. . . 4 ⊢ ((y ∈ A ∧ z ∈ B) → ∪ ran {⟨y, z⟩} ∈ B)
7	6	rgen2 2399	. . 3 ⊢ ∀y ∈ A ∀z ∈ B ∪ ran {⟨y, z⟩} ∈ B
8		sneq 3378	. . . . . . 7 ⊢ (x = ⟨y, z⟩ → {x} = {⟨y, z⟩})
9	8	rneqd 4506	. . . . . 6 ⊢ (x = ⟨y, z⟩ → ran {x} = ran {⟨y, z⟩})
10	9	unieqd 3582	. . . . 5 ⊢ (x = ⟨y, z⟩ → ∪ ran {x} = ∪ ran {⟨y, z⟩})
11	10	eleq1d 2103	. . . 4 ⊢ (x = ⟨y, z⟩ → (∪ ran {x} ∈ B ↔ ∪ ran {⟨y, z⟩} ∈ B))
12	11	ralxp 4422	. . 3 ⊢ (∀x ∈ (A × B)∪ ran {x} ∈ B ↔ ∀y ∈ A ∀z ∈ B ∪ ran {⟨y, z⟩} ∈ B)
13	7, 12	mpbir 134	. 2 ⊢ ∀x ∈ (A × B)∪ ran {x} ∈ B
14		df-2nd 5710	. . . . 5 ⊢ 2nd = (x ∈ V ↦ ∪ ran {x})
15	14	reseq1i 4551	. . . 4 ⊢ (2nd ↾ (A × B)) = ((x ∈ V ↦ ∪ ran {x}) ↾ (A × B))
16		ssv 2959	. . . . 5 ⊢ (A × B) ⊆ V
17		resmpt 4599	. . . . 5 ⊢ ((A × B) ⊆ V → ((x ∈ V ↦ ∪ ran {x}) ↾ (A × B)) = (x ∈ (A × B) ↦ ∪ ran {x}))
18	16, 17	ax-mp 7	. . . 4 ⊢ ((x ∈ V ↦ ∪ ran {x}) ↾ (A × B)) = (x ∈ (A × B) ↦ ∪ ran {x})
19	15, 18	eqtri 2057	. . 3 ⊢ (2nd ↾ (A × B)) = (x ∈ (A × B) ↦ ∪ ran {x})
20	19	fmpt 5262	. 2 ⊢ (∀x ∈ (A × B)∪ ran {x} ∈ B ↔ (2nd ↾ (A × B)):(A × B)⟶B)
21	13, 20	mpbi 133	1 ⊢ (2nd ↾ (A × B)):(A × B)⟶B

Syntax hints:   = wceq 1242   ∈ wcel 1390  ∀wral 2300  Vcvv 2551   ⊆ wss 2911  {csn 3367  ⟨cop 3370  ∪ cuni 3571   ↦ cmpt 3809   × cxp 4286  ran crn 4289   ↾ cres 4290  ⟶wf 4841  2^nd c2nd 5708

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-bndl 1396  ax-4 1397  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

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-fv 4853  df-2nd 5710

This theorem is referenced by:  fo2ndresm  5731  2ndcof  5733  f2ndf  5789