Theorem strfvssn 12700

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 12698	. 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁))
5	3	elexd 2776	. . 3 ⊢ (𝜑 → 𝑁 ∈ V)
6		fvssunirng 5573	. . 3 ⊢ (𝑁 ∈ V → (𝑆‘𝑁) ⊆ ∪ ran 𝑆)
7	5, 6	syl 14	. 2 ⊢ (𝜑 → (𝑆‘𝑁) ⊆ ∪ ran 𝑆)
8	4, 7	eqsstrd 3219	1 ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ⊆ wss 3157  ∪ cuni 3839  ran crn 4664  ‘cfv 5258  ℕcn 8990  Slot cslot 12677

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fv 5266  df-slot 12682

This theorem is referenced by: (None)