Theorem 2ndvalg 6198

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
2ndvalg	⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})

Proof of Theorem 2ndvalg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snexg 4214	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		rnexg 4928	. . 3 ⊢ ({𝐴} ∈ V → ran {𝐴} ∈ V)
3		uniexg 4471	. . 3 ⊢ (ran {𝐴} ∈ V → ∪ ran {𝐴} ∈ V)
4	1, 2, 3	3syl 17	. 2 ⊢ (𝐴 ∈ V → ∪ ran {𝐴} ∈ V)
5		sneq 3630	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
6	5	rneqd 4892	. . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
7	6	unieqd 3847	. . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴})
8		df-2nd 6196	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
9	7, 8	fvmptg 5634	. 2 ⊢ ((𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V) → (2nd ‘𝐴) = ∪ ran {𝐴})
10	4, 9	mpdan 421	1 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  Vcvv 2760  {csn 3619  ∪ cuni 3836  ran crn 4661  ‘cfv 5255  2^nd c2nd 6194

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fv 5263  df-2nd 6196

This theorem is referenced by:  2nd0  6200  op2nd  6202  elxp6  6224