Theorem strfvss 17148

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 17146	. . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝑆‘𝑁))
4		fvssunirn 6865	. . 3 ⊢ (𝑆‘𝑁) ⊆ ∪ ran 𝑆
5	3, 4	eqsstrdi 3967	. 2 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)
6		fvprc 6826	. . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅)
7		0ss 4341	. . 3 ⊢ ∅ ⊆ ∪ ran 𝑆
8	6, 7	eqsstrdi 3967	. 2 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)
9	5, 8	pm2.61i 182	1 ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  ∪ cuni 4851  ran crn 5625  ‘cfv 6492  Slot cslot 17142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-slot 17143

This theorem is referenced by:  wunstr  17149  prdsvallem  17408  prdsval  17409  prdsbas  17411  prdsplusg  17412  prdsmulr  17413  prdsvsca  17414  prdshom  17421