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Mirrors > Home > MPE Home > Th. List > strfv2d | Structured version Visualization version GIF version |
Description: Deduction version of strfv2 17175. (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 17157 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
4 | cnvcnv2 6199 | . . . . 5 ⊢ ◡◡𝑆 = (𝑆 ↾ V) | |
5 | 4 | fveq1i 6897 | . . . 4 ⊢ (◡◡𝑆‘(𝐸‘ndx)) = ((𝑆 ↾ V)‘(𝐸‘ndx)) |
6 | fvex 6909 | . . . . 5 ⊢ (𝐸‘ndx) ∈ V | |
7 | fvres 6915 | . . . . 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 2753 | . . 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 3483 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ V) |
14 | opelxpi 5715 | . . . . . . 7 ⊢ (((𝐸‘ndx) ∈ V ∧ 𝐶 ∈ V) → 〈(𝐸‘ndx), 𝐶〉 ∈ (V × V)) | |
15 | 6, 13, 14 | sylancr 585 | . . . . . 6 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (V × V)) |
16 | 11, 15 | elind 4192 | . . . . 5 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (𝑆 ∩ (V × V))) |
17 | cnvcnv 6198 | . . . . 5 ⊢ ◡◡𝑆 = (𝑆 ∩ (V × V)) | |
18 | 16, 17 | eleqtrrdi 2836 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ ◡◡𝑆) |
19 | funopfv 6948 | . . . 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 | eqtr3id 2779 | . 2 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶) |
22 | 3, 21 | eqtr2d 2766 | 1 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∩ cin 3943 〈cop 4636 × cxp 5676 ◡ccnv 5677 ↾ cres 5680 Fun wfun 6543 ‘cfv 6549 Slot cslot 17153 ndxcnx 17165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-res 5690 df-iota 6501 df-fun 6551 df-fv 6557 df-slot 17154 |
This theorem is referenced by: strfv2 17175 opelstrbas 17197 ebtwntg 28865 ecgrtg 28866 elntg 28867 edgfiedgval 28902 |
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