Theorem f2ndres 6332

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 2806	. . . . . . . 8 ⊢ 𝑦 ∈ V
2		vex 2806	. . . . . . . 8 ⊢ 𝑧 ∈ V
3	1, 2	op2nda 5228	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑧⟩} = 𝑧
4	3	eleq1i 2297	. . . . . 6 ⊢ (∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵 ↔ 𝑧 ∈ 𝐵)
5	4	biimpri 133	. . . . 5 ⊢ (𝑧 ∈ 𝐵 → ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
6	5	adantl 277	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
7	6	rgen2 2619	. . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵
8		sneq 3684	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
9	8	rneqd 4967	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
10	9	unieqd 3909	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑧⟩})
11	10	eleq1d 2300	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (∪ ran {𝑥} ∈ 𝐵 ↔ ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵))
12	11	ralxp 4879	. . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
13	7, 12	mpbir 146	. 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵
14		df-2nd 6313	. . . . 5 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
15	14	reseq1i 5015	. . . 4 ⊢ (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵))
16		ssv 3250	. . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V
17		resmpt 5067	. . . . 5 ⊢ ((𝐴 × 𝐵) ⊆ V → ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥}))
18	16, 17	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥})
19	15, 18	eqtri 2252	. . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥})
20	19	fmpt 5805	. 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵)
21	13, 20	mpbi 145	1 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵

Syntax hints:   = wceq 1398   ∈ wcel 2202  ∀wral 2511  Vcvv 2803   ⊆ wss 3201  {csn 3673  ⟨cop 3676  ∪ cuni 3898   ↦ cmpt 4155   × cxp 4729  ran crn 4732   ↾ cres 4733  ⟶wf 5329  2^nd c2nd 6311

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-2nd 6313

This theorem is referenced by:  fo2ndresm  6334  2ndcof  6336  f2ndf  6400  eucalgcvga  12691  tx2cn  15061