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Mirrors > Home > MPE Home > Th. List > strssd | Structured version Visualization version GIF version |
Description: Deduction version of strss 17136. (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 3982 | . . 3 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑇) |
7 | 1, 2, 3, 6 | strfvd 17130 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑇)) |
8 | 2, 4 | ssexd 5323 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
9 | funss 6564 | . . . 4 ⊢ (𝑆 ⊆ 𝑇 → (Fun 𝑇 → Fun 𝑆)) | |
10 | 4, 3, 9 | sylc 65 | . . 3 ⊢ (𝜑 → Fun 𝑆) |
11 | 1, 8, 10, 5 | strfvd 17130 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
12 | 7, 11 | eqtr3d 2775 | 1 ⊢ (𝜑 → (𝐸‘𝑇) = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3947 〈cop 4633 Fun wfun 6534 ‘cfv 6540 Slot cslot 17110 ndxcnx 17122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-slot 17111 |
This theorem is referenced by: strss 17136 |
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