Theorem strfvssn 12502

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.) (Revised by Jim Kingdon, 19-Jan-2023.)

Hypotheses

Ref	Expression
strfvssn.c	⊢ 𝐸 = Slot 𝑁
strfvssn.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
strfvssn.n	⊢ (𝜑 → 𝑁 ∈ ℕ)

Assertion

Ref	Expression
strfvssn	⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)

Proof of Theorem strfvssn

Step	Hyp	Ref	Expression
1		strfvssn.c	. . 3 ⊢ 𝐸 = Slot 𝑁
2		strfvssn.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
3		strfvssn.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	1, 2, 3	strnfvnd 12500	. 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁))
5	3	elexd 2765	. . 3 ⊢ (𝜑 → 𝑁 ∈ V)
6		fvssunirng 5545	. . 3 ⊢ (𝑁 ∈ V → (𝑆‘𝑁) ⊆ ∪ ran 𝑆)
7	5, 6	syl 14	. 2 ⊢ (𝜑 → (𝑆‘𝑁) ⊆ ∪ ran 𝑆)
8	4, 7	eqsstrd 3206	1 ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2160  Vcvv 2752   ⊆ wss 3144  ∪ cuni 3824  ran crn 4642  ‘cfv 5231  ℕcn 8937  Slot cslot 12479

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fv 5239  df-slot 12484

This theorem is referenced by: (None)