Theorem strfvssn 12497

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 12495	. 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁))
5	3	elexd 2762	. . 3 ⊢ (𝜑 → 𝑁 ∈ V)
6		fvssunirng 5542	. . 3 ⊢ (𝑁 ∈ V → (𝑆‘𝑁) ⊆ ∪ ran 𝑆)
7	5, 6	syl 14	. 2 ⊢ (𝜑 → (𝑆‘𝑁) ⊆ ∪ ran 𝑆)
8	4, 7	eqsstrd 3203	1 ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)

Syntax hints:   → wi 4   = wceq 1363   ∈ wcel 2158  Vcvv 2749   ⊆ wss 3141  ∪ cuni 3821  ran crn 4639  ‘cfv 5228  ℕcn 8932  Slot cslot 12474

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fv 5236  df-slot 12479

This theorem is referenced by: (None)