Theorem strfvss 17164

Description: A structure component extractor produces a value which is contained in a set dependent on 𝑆, but not 𝐸. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.)

Hypothesis

Ref	Expression
strfvss.e	⊢ 𝐸 = Slot 𝑁

Assertion

Ref	Expression
strfvss	⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆

Proof of Theorem strfvss

Step	Hyp	Ref	Expression
1		strfvss.e	. . . 4 ⊢ 𝐸 = Slot 𝑁
2		id 22	. . . 4 ⊢ (𝑆 ∈ V → 𝑆 ∈ V)
3	1, 2	strfvnd 17162	. . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝑆‘𝑁))
4		fvssunirn 6894	. . 3 ⊢ (𝑆‘𝑁) ⊆ ∪ ran 𝑆
5	3, 4	eqsstrdi 3994	. 2 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)
6		fvprc 6853	. . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅)
7		0ss 4366	. . 3 ⊢ ∅ ⊆ ∪ ran 𝑆
8	6, 7	eqsstrdi 3994	. 2 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)
9	5, 8	pm2.61i 182	1 ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  ∪ cuni 4874  ran crn 5642  ‘cfv 6514  Slot cslot 17158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fv 6522  df-slot 17159

This theorem is referenced by:  wunstr  17165  prdsvallem  17424  prdsval  17425  prdsbas  17427  prdsplusg  17428  prdsmulr  17429  prdsvsca  17430  prdshom  17437