Theorem f2ndres 7960

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 3434	. . . . . . . 8 ⊢ 𝑦 ∈ V
2		vex 3434	. . . . . . . 8 ⊢ 𝑧 ∈ V
3	1, 2	op2nda 6186	. . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑧⟩} = 𝑧
4	3	eleq1i 2828	. . . . . 6 ⊢ (∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵 ↔ 𝑧 ∈ 𝐵)
5	4	biimpri 228	. . . . 5 ⊢ (𝑧 ∈ 𝐵 → ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
6	5	adantl 481	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
7	6	rgen2 3178	. . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵
8		sneq 4578	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
9	8	rneqd 5887	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
10	9	unieqd 4864	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑧⟩})
11	10	eleq1d 2822	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (∪ ran {𝑥} ∈ 𝐵 ↔ ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵))
12	11	ralxp 5790	. . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
13	7, 12	mpbir 231	. 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵
14		df-2nd 7936	. . . . 5 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
15	14	reseq1i 5934	. . . 4 ⊢ (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵))
16		ssv 3947	. . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V
17		resmpt 5996	. . . . 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 2760	. . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥})
20	19	fmpt 7056	. 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵)
21	13, 20	mpbi 230	1 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890  {csn 4568  ⟨cop 4574  ∪ cuni 4851   ↦ cmpt 5167   × cxp 5622  ran crn 5625   ↾ cres 5626  ⟶wf 6488  2^nd c2nd 7934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-fun 6494  df-fn 6495  df-f 6496  df-2nd 7936

This theorem is referenced by:  fo2ndres  7962  2ndcof  7966  fparlem2  8056  f2ndf  8063  eucalgcvga  16546  2ndfcl  18155  gaid  19265  tx2cn  23585  txkgen  23627  xpinpreima  34066  xpinpreima2  34067  2ndmbfm  34421  filnetlem4  36579  hausgraph  43651