Theorem 2ndvalg 6336

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 4296	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		rnexg 5021	. . 3 ⊢ ({𝐴} ∈ V → ran {𝐴} ∈ V)
3		uniexg 4559	. . 3 ⊢ (ran {𝐴} ∈ V → ∪ ran {𝐴} ∈ V)
4	1, 2, 3	3syl 17	. 2 ⊢ (𝐴 ∈ V → ∪ ran {𝐴} ∈ V)
5		sneq 3699	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
6	5	rneqd 4985	. . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
7	6	unieqd 3924	. . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴})
8		df-2nd 6334	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
9	7, 8	fvmptg 5752	. 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 1398   ∈ wcel 2203  Vcvv 2812  {csn 3688  ∪ cuni 3913  ran crn 4749  ‘cfv 5351  2^nd c2nd 6332

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-iota 5311  df-fun 5353  df-fv 5359  df-2nd 6334

This theorem is referenced by:  2nd0  6338  op2nd  6340  elxp6  6362