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| Description: Deduction version of strfv2 17240. (Contributed by Mario Carneiro, 30-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| strfv2d.e | ⊢ 𝐸 = Slot (𝐸‘ndx) | 
| strfv2d.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) | 
| strfv2d.f | ⊢ (𝜑 → Fun ◡◡𝑆) | 
| strfv2d.n | ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) | 
| strfv2d.c | ⊢ (𝜑 → 𝐶 ∈ 𝑊) | 
| Ref | Expression | 
|---|---|
| strfv2d | ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | strfv2d.e | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
| 2 | strfv2d.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
| 3 | 1, 2 | strfvnd 17223 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) | 
| 4 | cnvcnv2 6212 | . . . . 5 ⊢ ◡◡𝑆 = (𝑆 ↾ V) | |
| 5 | 4 | fveq1i 6906 | . . . 4 ⊢ (◡◡𝑆‘(𝐸‘ndx)) = ((𝑆 ↾ V)‘(𝐸‘ndx)) | 
| 6 | fvex 6918 | . . . . 5 ⊢ (𝐸‘ndx) ∈ V | |
| 7 | fvres 6924 | . . . . 5 ⊢ ((𝐸‘ndx) ∈ V → ((𝑆 ↾ V)‘(𝐸‘ndx)) = (𝑆‘(𝐸‘ndx))) | |
| 8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ ((𝑆 ↾ V)‘(𝐸‘ndx)) = (𝑆‘(𝐸‘ndx)) | 
| 9 | 5, 8 | eqtri 2764 | . . 3 ⊢ (◡◡𝑆‘(𝐸‘ndx)) = (𝑆‘(𝐸‘ndx)) | 
| 10 | strfv2d.f | . . . 4 ⊢ (𝜑 → Fun ◡◡𝑆) | |
| 11 | strfv2d.n | . . . . . 6 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) | |
| 12 | strfv2d.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
| 13 | 12 | elexd 3503 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ V) | 
| 14 | opelxpi 5721 | . . . . . . 7 ⊢ (((𝐸‘ndx) ∈ V ∧ 𝐶 ∈ V) → 〈(𝐸‘ndx), 𝐶〉 ∈ (V × V)) | |
| 15 | 6, 13, 14 | sylancr 587 | . . . . . 6 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (V × V)) | 
| 16 | 11, 15 | elind 4199 | . . . . 5 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (𝑆 ∩ (V × V))) | 
| 17 | cnvcnv 6211 | . . . . 5 ⊢ ◡◡𝑆 = (𝑆 ∩ (V × V)) | |
| 18 | 16, 17 | eleqtrrdi 2851 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ ◡◡𝑆) | 
| 19 | funopfv 6957 | . . . 4 ⊢ (Fun ◡◡𝑆 → (〈(𝐸‘ndx), 𝐶〉 ∈ ◡◡𝑆 → (◡◡𝑆‘(𝐸‘ndx)) = 𝐶)) | |
| 20 | 10, 18, 19 | sylc 65 | . . 3 ⊢ (𝜑 → (◡◡𝑆‘(𝐸‘ndx)) = 𝐶) | 
| 21 | 9, 20 | eqtr3id 2790 | . 2 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶) | 
| 22 | 3, 21 | eqtr2d 2777 | 1 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∩ cin 3949 〈cop 4631 × cxp 5682 ◡ccnv 5683 ↾ cres 5686 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: strfv2 17240 opelstrbas 17261 ebtwntg 28998 ecgrtg 28999 elntg 29000 edgfiedgval 29035 | 
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