Theorem 2ndval 7974

Description: The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
2ndval	⊢ (2nd ‘𝐴) = ∪ ran {𝐴}

Proof of Theorem 2ndval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4637	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	rneqd 5935	. . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
3	2	unieqd 4921	. . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴})
4		df-2nd 7972	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
5		snex 5430	. . . . 5 ⊢ {𝐴} ∈ V
6	5	rnex 7899	. . . 4 ⊢ ran {𝐴} ∈ V
7	6	uniex 7727	. . 3 ⊢ ∪ ran {𝐴} ∈ V
8	3, 4, 7	fvmpt 6995	. 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})
9		fvprc 6880	. . 3 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∅)
10		snprc 4720	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11	10	biimpi 215	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
12	11	rneqd 5935	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ran ∅)
13		rn0 5923	. . . . . 6 ⊢ ran ∅ = ∅
14	12, 13	eqtrdi 2788	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ∅)
15	14	unieqd 4921	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∪ ∅)
16		uni0 4938	. . . 4 ⊢ ∪ ∅ = ∅
17	15, 16	eqtrdi 2788	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∅)
18	9, 17	eqtr4d 2775	. 2 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})
19	8, 18	pm2.61i 182	1 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ∅c0 4321  {csn 4627  ∪ cuni 4907  ran crn 5676  ‘cfv 6540  2^nd c2nd 7970

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fv 6548  df-2nd 7972

This theorem is referenced by:  2ndnpr  7976  2nd0  7978  op2nd  7980  2nd2val  8000  elxp6  8005