Theorem 2ndval 4072

Description: The value of the function that extracts the second member of an ordered pair.

Assertion

Ref	Expression
2ndval	⊢ (2nd ‘A) = ∪ran { A}

Proof of Theorem 2ndval

Step	Hyp	Ref	Expression
1		snex 2745	. . . . 5 ⊢ {A} ∈ V
2		rnexg 3353	. . . . 5 ⊢ ({A} ∈ V → ran { A} ∈ V)
3	1, 2	ax-mp 7	. . . 4 ⊢ ran { A} ∈ V
4	3	uniex 2865	. . 3 ⊢ ∪ran { A} ∈ V
5		sneq 2413	. . . . . . 7 ⊢ (x = A → {x} = {A})
6	5	rneqd 3336	. . . . . 6 ⊢ (x = A → ran { x} = ran { A})
7	6	unieqd 2507	. . . . 5 ⊢ (x = A → ∪ran { x} = ∪ran { A})
8	7	fvopabg 3776	. . . 4 ⊢ ((A ∈ V ⋀ ∪ran { A} ∈ V) → ({⟨x, y⟩∣y = ∪ran { x}} ‘A) = ∪ran { A})
9		df-2nd 4070	. . . . 5 ⊢ 2nd = {⟨x, y⟩∣y = ∪ran { x}}
10	9	fveq1i 3716	. . . 4 ⊢ (2nd ‘A) = ({⟨x, y⟩∣y = ∪ran { x}} ‘A)
11	8, 10	syl5eq 1516	. . 3 ⊢ ((A ∈ V ⋀ ∪ran { A} ∈ V) → (2nd ‘A) = ∪ran { A})
12	4, 11	mpan2 695	. 2 ⊢ (A ∈ V → (2nd ‘A) = ∪ran { A})
13		fvprc 3712	. . 3 ⊢ (¬ A ∈ V → (2nd ‘A) = ∅)
14		snprc 2439	. . . . . . . 8 ⊢ (¬ A ∈ V ↔ {A} = ∅)
15	14	biimp 151	. . . . . . 7 ⊢ (¬ A ∈ V → {A} = ∅)
16	15	rneqd 3336	. . . . . 6 ⊢ (¬ A ∈ V → ran { A} = ran ∅)
17		rn0 3349	. . . . . 6 ⊢ ran ∅ = ∅
18	16, 17	syl6eq 1520	. . . . 5 ⊢ (¬ A ∈ V → ran { A} = ∅)
19	18	unieqd 2507	. . . 4 ⊢ (¬ A ∈ V → ∪ran { A} = ∪∅)
20		uni0 2520	. . . 4 ⊢ ∪∅ = ∅
21	19, 20	syl6eq 1520	. . 3 ⊢ (¬ A ∈ V → ∪ran { A} = ∅)
22	13, 21	eqtr4d 1507	. 2 ⊢ (¬ A ∈ V → (2nd ‘A) = ∪ran { A})
23	12, 22	pm2.61i 126	1 ⊢ (2nd ‘A) = ∪ran { A}

Syntax hints:  ¬ wn 2   ⋀ wa 223   = wceq 954   ∈ wcel 956  Vcvv 1807  ∅c0 2276  {csn 2405  ∪cuni 2498  {copab 2661  ran crn 3166   ‘cfv 3177  2^nd c2nd 4068

This theorem is referenced by:  2nd0 4074  op2nd 4076  2nd2val 4086  elxp6 4092

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fv 3193  df-2nd 4070

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