Theorem 1stvalg 6251

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 4244	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		dmexg 4961	. . 3 ⊢ ({𝐴} ∈ V → dom {𝐴} ∈ V)
3		uniexg 4504	. . 3 ⊢ (dom {𝐴} ∈ V → ∪ dom {𝐴} ∈ V)
4	1, 2, 3	3syl 17	. 2 ⊢ (𝐴 ∈ V → ∪ dom {𝐴} ∈ V)
5		sneq 3654	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
6	5	dmeqd 4899	. . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
7	6	unieqd 3875	. . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴})
8		df-1st 6249	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
9	7, 8	fvmptg 5678	. 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 2178  Vcvv 2776  {csn 3643  ∪ cuni 3864  dom cdm 4693  ‘cfv 5290  1^st c1st 6247

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fv 5298  df-1st 6249

This theorem is referenced by:  1st0  6253  op1st  6255  elxp6  6278