Theorem strfvssn 12416

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 12414	. 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁))
5	3	elexd 2739	. . 3 ⊢ (𝜑 → 𝑁 ∈ V)
6		fvssunirng 5501	. . 3 ⊢ (𝑁 ∈ V → (𝑆‘𝑁) ⊆ ∪ ran 𝑆)
7	5, 6	syl 14	. 2 ⊢ (𝜑 → (𝑆‘𝑁) ⊆ ∪ ran 𝑆)
8	4, 7	eqsstrd 3178	1 ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)

Syntax hints:   → wi 4   = wceq 1343   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  ∪ cuni 3789  ran crn 4605  ‘cfv 5188  ℕcn 8857  Slot cslot 12393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fv 5196  df-slot 12398

This theorem is referenced by: (None)