Theorem strfvssn 13324

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 13322	. 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁))
5	3	elexd 2829	. . 3 ⊢ (𝜑 → 𝑁 ∈ V)
6		fvssunirng 5692	. . 3 ⊢ (𝑁 ∈ V → (𝑆‘𝑁) ⊆ ∪ ran 𝑆)
7	5, 6	syl 14	. 2 ⊢ (𝜑 → (𝑆‘𝑁) ⊆ ∪ ran 𝑆)
8	4, 7	eqsstrd 3278	1 ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ⊆ wss 3214  ∪ cuni 3920  ran crn 4757  ‘cfv 5359  ℕcn 9259  Slot cslot 13301

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-iota 5319  df-fun 5361  df-fv 5367  df-slot 13306

This theorem is referenced by:  prdsvallem  13570  prdsval  14122