Theorem f2ndres 6353

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 2815	. . . . . . . 8 ⊢ 𝑦 ∈ V
2		vex 2815	. . . . . . . 8 ⊢ 𝑧 ∈ V
3	1, 2	op2nda 5246	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑧⟩} = 𝑧
4	3	eleq1i 2298	. . . . . 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 2628	. . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵
8		sneq 3699	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
9	8	rneqd 4985	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
10	9	unieqd 3924	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑧⟩})
11	10	eleq1d 2301	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (∪ ran {𝑥} ∈ 𝐵 ↔ ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵))
12	11	ralxp 4897	. . 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 6334	. . . . 5 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
15	14	reseq1i 5033	. . . 4 ⊢ (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵))
16		ssv 3259	. . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V
17		resmpt 5085	. . . . 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 2253	. . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥})
20	19	fmpt 5826	. 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 2203  ∀wral 2520  Vcvv 2812   ⊆ wss 3210  {csn 3688  ⟨cop 3691  ∪ cuni 3913   ↦ cmpt 4170   × cxp 4746  ran crn 4749   ↾ cres 4750  ⟶wf 5347  2^nd c2nd 6332

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 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-2nd 6334

This theorem is referenced by:  fo2ndresm  6355  2ndcof  6357  f2ndf  6421  eucalgcvga  12751  tx2cn  15127