Theorem strfv2 17163

Description: A variation on strfv 17164 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 22	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ 𝑉)
9	1, 3, 5, 7, 8	strfv2d 17162	1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574  ^◡ccnv 5623  Fun wfun 6486  ‘cfv 6492  Slot cslot 17142  ndxcnx 17154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-res 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-slot 17143

This theorem is referenced by:  strfv  17164