Theorem strfvss 17157

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 17155	. . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝑆‘𝑁))
4		fvssunirn 6891	. . 3 ⊢ (𝑆‘𝑁) ⊆ ∪ ran 𝑆
5	3, 4	eqsstrdi 3991	. 2 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)
6		fvprc 6850	. . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅)
7		0ss 4363	. . 3 ⊢ ∅ ⊆ ∪ ran 𝑆
8	6, 7	eqsstrdi 3991	. 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 3447   ⊆ wss 3914  ∅c0 4296  ∪ cuni 4871  ran crn 5639  ‘cfv 6511  Slot cslot 17151

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fv 6519  df-slot 17152

This theorem is referenced by:  wunstr  17158  prdsvallem  17417  prdsval  17418  prdsbas  17420  prdsplusg  17421  prdsmulr  17422  prdsvsca  17423  prdshom  17430