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| Description: Deduction version of strss 17244. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| strssd.e | ⊢ 𝐸 = Slot (𝐸‘ndx) | 
| strssd.t | ⊢ (𝜑 → 𝑇 ∈ 𝑉) | 
| strssd.f | ⊢ (𝜑 → Fun 𝑇) | 
| strssd.s | ⊢ (𝜑 → 𝑆 ⊆ 𝑇) | 
| strssd.n | ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) | 
| Ref | Expression | 
|---|---|
| strssd | ⊢ (𝜑 → (𝐸‘𝑇) = (𝐸‘𝑆)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | strssd.e | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
| 2 | strssd.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
| 3 | strssd.f | . . 3 ⊢ (𝜑 → Fun 𝑇) | |
| 4 | strssd.s | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑇) | |
| 5 | strssd.n | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) | |
| 6 | 4, 5 | sseldd 3983 | . . 3 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑇) | 
| 7 | 1, 2, 3, 6 | strfvd 17238 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑇)) | 
| 8 | 2, 4 | ssexd 5323 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) | 
| 9 | funss 6584 | . . . 4 ⊢ (𝑆 ⊆ 𝑇 → (Fun 𝑇 → Fun 𝑆)) | |
| 10 | 4, 3, 9 | sylc 65 | . . 3 ⊢ (𝜑 → Fun 𝑆) | 
| 11 | 1, 8, 10, 5 | strfvd 17238 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) | 
| 12 | 7, 11 | eqtr3d 2778 | 1 ⊢ (𝜑 → (𝐸‘𝑇) = (𝐸‘𝑆)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 〈cop 4631 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-iota 6513 df-fun 6562 df-fv 6568 df-slot 17220 | 
| This theorem is referenced by: strss 17244 | 
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