Theorem f2ndres 7997

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 3479	. . . . . . . 8 ⊢ 𝑦 ∈ V
2		vex 3479	. . . . . . . 8 ⊢ 𝑧 ∈ V
3	1, 2	op2nda 6225	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑧⟩} = 𝑧
4	3	eleq1i 2825	. . . . . 6 ⊢ (∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵 ↔ 𝑧 ∈ 𝐵)
5	4	biimpri 227	. . . . 5 ⊢ (𝑧 ∈ 𝐵 → ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
6	5	adantl 483	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
7	6	rgen2 3198	. . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵
8		sneq 4638	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
9	8	rneqd 5936	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
10	9	unieqd 4922	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑧⟩})
11	10	eleq1d 2819	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (∪ ran {𝑥} ∈ 𝐵 ↔ ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵))
12	11	ralxp 5840	. . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
13	7, 12	mpbir 230	. 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵
14		df-2nd 7973	. . . . 5 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
15	14	reseq1i 5976	. . . 4 ⊢ (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵))
16		ssv 4006	. . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V
17		resmpt 6036	. . . . 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 2761	. . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥})
20	19	fmpt 7107	. 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵)
21	13, 20	mpbi 229	1 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵

Syntax hints:   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ⊆ wss 3948  {csn 4628  ⟨cop 4634  ∪ cuni 4908   ↦ cmpt 5231   × cxp 5674  ran crn 5677   ↾ cres 5678  ⟶wf 6537  2^nd c2nd 7971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6543  df-fn 6544  df-f 6545  df-2nd 7973

This theorem is referenced by:  fo2ndres  7999  2ndcof  8003  fparlem2  8096  f2ndf  8103  eucalgcvga  16520  2ndfcl  18147  gaid  19158  tx2cn  23106  txkgen  23148  xpinpreima  32875  xpinpreima2  32876  2ndmbfm  33249  filnetlem4  35255  hausgraph  41940