Theorem 2ndvalg 6371

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 4319	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		rnexg 5045	. . 3 ⊢ ({𝐴} ∈ V → ran {𝐴} ∈ V)
3		uniexg 4583	. . 3 ⊢ (ran {𝐴} ∈ V → ∪ ran {𝐴} ∈ V)
4	1, 2, 3	3syl 17	. 2 ⊢ (𝐴 ∈ V → ∪ ran {𝐴} ∈ V)
5		sneq 3719	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
6	5	rneqd 5009	. . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
7	6	unieqd 3944	. . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴})
8		df-2nd 6369	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
9	7, 8	fvmptg 5778	. 2 ⊢ ((𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V) → (2nd ‘𝐴) = ∪ ran {𝐴})
10	4, 9	mpdan 425	1 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821  {csn 3708  ∪ cuni 3933  ran crn 4773  ‘cfv 5375  2^nd c2nd 6367

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fv 5383  df-2nd 6369

This theorem is referenced by:  2nd0  6373  op2nd  6375  elxp6  6397