Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > strssd | Structured version Visualization version GIF version |
Description: Deduction version of strss 15910. (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 3604 | . . 3 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑇) |
7 | 1, 2, 3, 6 | strfvd 15904 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑇)) |
8 | 2, 4 | ssexd 4805 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
9 | funss 5907 | . . . 4 ⊢ (𝑆 ⊆ 𝑇 → (Fun 𝑇 → Fun 𝑆)) | |
10 | 4, 3, 9 | sylc 65 | . . 3 ⊢ (𝜑 → Fun 𝑆) |
11 | 1, 8, 10, 5 | strfvd 15904 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
12 | 7, 11 | eqtr3d 2658 | 1 ⊢ (𝜑 → (𝐸‘𝑇) = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 〈cop 4183 Fun wfun 5882 ‘cfv 5888 ndxcnx 15854 Slot cslot 15856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-slot 15861 |
This theorem is referenced by: strss 15910 |
Copyright terms: Public domain | W3C validator |