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| Description: A variation on strfv 17241 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.) | 
| Ref | Expression | 
|---|---|
| strfv2.s | ⊢ 𝑆 ∈ V | 
| strfv2.f | ⊢ Fun ◡◡𝑆 | 
| strfv2.e | ⊢ 𝐸 = Slot (𝐸‘ndx) | 
| strfv2.n | ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 | 
| Ref | Expression | 
|---|---|
| strfv2 | ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) | 
| 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 17239 | 1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 〈cop 4631 ◡ccnv 5683 Fun wfun 6554 ‘cfv 6560 Slot cslot 17219 ndxcnx 17231 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-res 5696 df-iota 6513 df-fun 6562 df-fv 6568 df-slot 17220 | 
| This theorem is referenced by: strfv 17241 | 
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