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Mirrors > Home > MPE Home > Th. List > strssd | Structured version Visualization version GIF version |
Description: Deduction version of strss 16526. (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 3916 | . . 3 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑇) |
7 | 1, 2, 3, 6 | strfvd 16520 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑇)) |
8 | 2, 4 | ssexd 5192 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
9 | funss 6343 | . . . 4 ⊢ (𝑆 ⊆ 𝑇 → (Fun 𝑇 → Fun 𝑆)) | |
10 | 4, 3, 9 | sylc 65 | . . 3 ⊢ (𝜑 → Fun 𝑆) |
11 | 1, 8, 10, 5 | strfvd 16520 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
12 | 7, 11 | eqtr3d 2835 | 1 ⊢ (𝜑 → (𝐸‘𝑇) = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 〈cop 4531 Fun wfun 6318 ‘cfv 6324 ndxcnx 16472 Slot cslot 16474 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-slot 16479 |
This theorem is referenced by: strss 16526 |
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