Theorem 1stvalg 5711

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		snexgOLD 3926	. . 3 ⊢ (A ∈ V → {A} ∈ V)
2		dmexg 4539	. . 3 ⊢ ({A} ∈ V → dom {A} ∈ V)
3		uniexg 4141	. . 3 ⊢ (dom {A} ∈ V → ∪ dom {A} ∈ V)
4	1, 2, 3	3syl 17	. 2 ⊢ (A ∈ V → ∪ dom {A} ∈ V)
5		sneq 3378	. . . . 5 ⊢ (x = A → {x} = {A})
6	5	dmeqd 4480	. . . 4 ⊢ (x = A → dom {x} = dom {A})
7	6	unieqd 3582	. . 3 ⊢ (x = A → ∪ dom {x} = ∪ dom {A})
8		df-1st 5709	. . 3 ⊢ 1st = (x ∈ V ↦ ∪ dom {x})
9	7, 8	fvmptg 5191	. 2 ⊢ ((A ∈ V ∧ ∪ dom {A} ∈ V) → (1st ‘A) = ∪ dom {A})
10	4, 9	mpdan 398	1 ⊢ (A ∈ V → (1st ‘A) = ∪ dom {A})

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {csn 3367  ∪ cuni 3571  dom cdm 4288  ‘cfv 4845  1^st c1st 5707

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-1st 5709

This theorem is referenced by:  1st0  5713  op1st  5715  elxp6  5738