Theorem 1stvalg 6133

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 4179	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		dmexg 4884	. . 3 ⊢ ({𝐴} ∈ V → dom {𝐴} ∈ V)
3		uniexg 4433	. . 3 ⊢ (dom {𝐴} ∈ V → ∪ dom {𝐴} ∈ V)
4	1, 2, 3	3syl 17	. 2 ⊢ (𝐴 ∈ V → ∪ dom {𝐴} ∈ V)
5		sneq 3600	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
6	5	dmeqd 4822	. . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
7	6	unieqd 3816	. . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴})
8		df-1st 6131	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
9	7, 8	fvmptg 5584	. 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 1353   ∈ wcel 2146  Vcvv 2735  {csn 3589  ∪ cuni 3805  dom cdm 4620  ‘cfv 5208  1^st c1st 6129

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-iota 5170  df-fun 5210  df-fv 5216  df-1st 6131

This theorem is referenced by:  1st0  6135  op1st  6137  elxp6  6160