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Mirrors > Home > MPE Home > Th. List > strfv2d | Structured version Visualization version GIF version |
Description: Deduction version of strfv2 16522. (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 16494 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
4 | cnvcnv2 6017 | . . . . 5 ⊢ ◡◡𝑆 = (𝑆 ↾ V) | |
5 | 4 | fveq1i 6646 | . . . 4 ⊢ (◡◡𝑆‘(𝐸‘ndx)) = ((𝑆 ↾ V)‘(𝐸‘ndx)) |
6 | fvex 6658 | . . . . 5 ⊢ (𝐸‘ndx) ∈ V | |
7 | fvres 6664 | . . . . 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 2821 | . . 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 3461 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ V) |
14 | opelxpi 5556 | . . . . . . 7 ⊢ (((𝐸‘ndx) ∈ V ∧ 𝐶 ∈ V) → 〈(𝐸‘ndx), 𝐶〉 ∈ (V × V)) | |
15 | 6, 13, 14 | sylancr 590 | . . . . . 6 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (V × V)) |
16 | 11, 15 | elind 4121 | . . . . 5 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (𝑆 ∩ (V × V))) |
17 | cnvcnv 6016 | . . . . 5 ⊢ ◡◡𝑆 = (𝑆 ∩ (V × V)) | |
18 | 16, 17 | eleqtrrdi 2901 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ ◡◡𝑆) |
19 | funopfv 6692 | . . . 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 | syl5eqr 2847 | . 2 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶) |
22 | 3, 21 | eqtr2d 2834 | 1 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 〈cop 4531 × cxp 5517 ◡ccnv 5518 ↾ cres 5521 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-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-slot 16479 |
This theorem is referenced by: strfv2 16522 opelstrbas 16589 ebtwntg 26776 ecgrtg 26777 elntg 26778 edgfiedgval 26810 |
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