Theorem 2ndval 7993

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 4594	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	rneqd 5920	. . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
3	2	unieqd 4880	. . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴})
4		df-2nd 7991	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
5		snex 5397	. . . . 5 ⊢ {𝐴} ∈ V
6	5	rnex 7911	. . . 4 ⊢ ran {𝐴} ∈ V
7	6	uniex 7747	. . 3 ⊢ ∪ ran {𝐴} ∈ V
8	3, 4, 7	fvmpt 6985	. 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})
9		fvprc 6869	. . 3 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∅)
10		snprc 4678	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11	10	biimpi 219	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
12	11	rneqd 5920	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ran ∅)
13		rn0 5908	. . . . . 6 ⊢ ran ∅ = ∅
14	12, 13	eqtrdi 2812	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ∅)
15	14	unieqd 4880	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∪ ∅)
16		uni0 4896	. . . 4 ⊢ ∪ ∅ = ∅
17	15, 16	eqtrdi 2812	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∅)
18	9, 17	eqtr4d 2799	. 2 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})
19	8, 18	pm2.61i 184	1 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ∅c0 4279  {csn 4584  ∪ cuni 4867  ran crn 5652  ‘cfv 6531  2^nd c2nd 7989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6487  df-fun 6533  df-fv 6539  df-2nd 7991

This theorem is used by:  2ndnpr  7995  2nd0  7997  op2nd  7999  2nd2val  8019  elxp6  8024