Theorem 2ndval 7692

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 4577	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	rneqd 5808	. . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
3	2	unieqd 4852	. . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴})
4		df-2nd 7690	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
5		snex 5332	. . . . 5 ⊢ {𝐴} ∈ V
6	5	rnex 7617	. . . 4 ⊢ ran {𝐴} ∈ V
7	6	uniex 7467	. . 3 ⊢ ∪ ran {𝐴} ∈ V
8	3, 4, 7	fvmpt 6768	. 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})
9		fvprc 6663	. . 3 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∅)
10		snprc 4653	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11	10	biimpi 218	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
12	11	rneqd 5808	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ran ∅)
13		rn0 5796	. . . . . 6 ⊢ ran ∅ = ∅
14	12, 13	syl6eq 2872	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ∅)
15	14	unieqd 4852	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∪ ∅)
16		uni0 4866	. . . 4 ⊢ ∪ ∅ = ∅
17	15, 16	syl6eq 2872	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∅)
18	9, 17	eqtr4d 2859	. 2 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})
19	8, 18	pm2.61i 184	1 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ∅c0 4291  {csn 4567  ∪ cuni 4838  ran crn 5556  ‘cfv 6355  2^nd c2nd 7688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fv 6363  df-2nd 7690

This theorem is referenced by:  2ndnpr  7694  2nd0  7696  op2nd  7698  2nd2val  7718  elxp6  7723