Theorem 2ndvalg 6146

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 4186	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		rnexg 4894	. . 3 ⊢ ({𝐴} ∈ V → ran {𝐴} ∈ V)
3		uniexg 4441	. . 3 ⊢ (ran {𝐴} ∈ V → ∪ ran {𝐴} ∈ V)
4	1, 2, 3	3syl 17	. 2 ⊢ (𝐴 ∈ V → ∪ ran {𝐴} ∈ V)
5		sneq 3605	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
6	5	rneqd 4858	. . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
7	6	unieqd 3822	. . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴})
8		df-2nd 6144	. . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
9	7, 8	fvmptg 5594	. 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 1353   ∈ wcel 2148  Vcvv 2739  {csn 3594  ∪ cuni 3811  ran crn 4629  ‘cfv 5218  2^nd c2nd 6142

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fv 5226  df-2nd 6144

This theorem is referenced by:  2nd0  6148  op2nd  6150  elxp6  6172