Theorem strfv2 17360

Description: A variation on strfv 17361 to avoid asserting that 𝑆 itself is a function, which involves sethood of all the ordered pair components of 𝑆. (Contributed by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
strfv2.s	⊢ 𝑆 ∈ V
strfv2.f	⊢ Fun ◡◡𝑆
strfv2.e	⊢ 𝐸 = Slot (𝐸‘ndx)
strfv2.n	⊢ ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆

Assertion

Ref	Expression
strfv2	⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))

Proof of Theorem strfv2

Step	Hyp	Ref	Expression
1		strfv2.e	. 2 ⊢ 𝐸 = Slot (𝐸‘ndx)
2		strfv2.s	. . 3 ⊢ 𝑆 ∈ V
3	2	a1i 11	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝑆 ∈ V)
4		strfv2.f	. . 3 ⊢ Fun ◡◡𝑆
5	4	a1i 11	. 2 ⊢ (𝐶 ∈ 𝑉 → Fun ◡◡𝑆)
6		strfv2.n	. . 3 ⊢ ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆
7	6	a1i 11	. 2 ⊢ (𝐶 ∈ 𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
8		id 23	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ 𝑉)
9	1, 3, 5, 7, 8	strfv2d 17359	1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⟨cop 4590  ^◡ccnv 5650  Fun wfun 6525  ‘cfv 6531  Slot cslot 17339  ndxcnx 17351

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-iota 6487  df-fun 6533  df-fv 6539  df-slot 17340

This theorem is used by:  strfv  17361