Theorem 1stvalg 6228

Description: The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
1stvalg	⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})

Proof of Theorem 1stvalg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snexg 4228	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		dmexg 4942	. . 3 ⊢ ({𝐴} ∈ V → dom {𝐴} ∈ V)
3		uniexg 4486	. . 3 ⊢ (dom {𝐴} ∈ V → ∪ dom {𝐴} ∈ V)
4	1, 2, 3	3syl 17	. 2 ⊢ (𝐴 ∈ V → ∪ dom {𝐴} ∈ V)
5		sneq 3644	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
6	5	dmeqd 4880	. . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
7	6	unieqd 3861	. . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴})
8		df-1st 6226	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
9	7, 8	fvmptg 5655	. 2 ⊢ ((𝐴 ∈ V ∧ ∪ dom {𝐴} ∈ V) → (1st ‘𝐴) = ∪ dom {𝐴})
10	4, 9	mpdan 421	1 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2176  Vcvv 2772  {csn 3633  ∪ cuni 3850  dom cdm 4675  ‘cfv 5271  1^st c1st 6224

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fv 5279  df-1st 6226

This theorem is referenced by:  1st0  6230  op1st  6232  elxp6  6255