Theorem 2ndval 8033

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 4658	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	rneqd 5963	. . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
3	2	unieqd 4944	. . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴})
4		df-2nd 8031	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
5		snex 5451	. . . . 5 ⊢ {𝐴} ∈ V
6	5	rnex 7950	. . . 4 ⊢ ran {𝐴} ∈ V
7	6	uniex 7776	. . 3 ⊢ ∪ ran {𝐴} ∈ V
8	3, 4, 7	fvmpt 7029	. 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})
9		fvprc 6912	. . 3 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∅)
10		snprc 4742	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11	10	biimpi 216	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
12	11	rneqd 5963	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ran ∅)
13		rn0 5950	. . . . . 6 ⊢ ran ∅ = ∅
14	12, 13	eqtrdi 2796	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ∅)
15	14	unieqd 4944	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∪ ∅)
16		uni0 4959	. . . 4 ⊢ ∪ ∅ = ∅
17	15, 16	eqtrdi 2796	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∅)
18	9, 17	eqtr4d 2783	. 2 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})
19	8, 18	pm2.61i 182	1 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  {csn 4648  ∪ cuni 4931  ran crn 5701  ‘cfv 6573  2^nd c2nd 8029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-2nd 8031

This theorem is referenced by:  2ndnpr  8035  2nd0  8037  op2nd  8039  2nd2val  8059  elxp6  8064