Theorem 2ndvalg 5928

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 4025	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		rnexg 4711	. . 3 ⊢ ({𝐴} ∈ V → ran {𝐴} ∈ V)
3		uniexg 4275	. . 3 ⊢ (ran {𝐴} ∈ V → ∪ ran {𝐴} ∈ V)
4	1, 2, 3	3syl 17	. 2 ⊢ (𝐴 ∈ V → ∪ ran {𝐴} ∈ V)
5		sneq 3461	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
6	5	rneqd 4677	. . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
7	6	unieqd 3670	. . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴})
8		df-2nd 5926	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
9	7, 8	fvmptg 5393	. 2 ⊢ ((𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V) → (2nd ‘𝐴) = ∪ ran {𝐴})
10	4, 9	mpdan 413	1 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})

Syntax hints:   → wi 4   = wceq 1290   ∈ wcel 1439  Vcvv 2620  {csn 3450  ∪ cuni 3659  ran crn 4453  ‘cfv 5028  2^nd c2nd 5924

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-iota 4993  df-fun 5030  df-fv 5036  df-2nd 5926

This theorem is referenced by:  2nd0  5930  op2nd  5932  elxp6  5954