Theorem 2ndval 7171

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 4187	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	rneqd 5353	. . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
3	2	unieqd 4446	. . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴})
4		df-2nd 7169	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
5		snex 4908	. . . . 5 ⊢ {𝐴} ∈ V
6	5	rnex 7100	. . . 4 ⊢ ran {𝐴} ∈ V
7	6	uniex 6953	. . 3 ⊢ ∪ ran {𝐴} ∈ V
8	3, 4, 7	fvmpt 6282	. 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})
9		fvprc 6185	. . 3 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∅)
10		snprc 4253	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11	10	biimpi 206	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
12	11	rneqd 5353	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ran ∅)
13		rn0 5377	. . . . . 6 ⊢ ran ∅ = ∅
14	12, 13	syl6eq 2672	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ∅)
15	14	unieqd 4446	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∪ ∅)
16		uni0 4465	. . . 4 ⊢ ∪ ∅ = ∅
17	15, 16	syl6eq 2672	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∅)
18	9, 17	eqtr4d 2659	. 2 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})
19	8, 18	pm2.61i 176	1 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  {csn 4177  ∪ cuni 4436  ran crn 5115  ‘cfv 5888  2^nd c2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-2nd 7169

This theorem is referenced by:  2ndnpr  7173  2nd0  7175  op2nd  7177  2nd2val  7195  elxp6  7200