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Mirrors > Home > MPE Home > Th. List > strfv2d | Structured version Visualization version GIF version |
Description: Deduction version of strfv2 16532. (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 16504 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
4 | cnvcnv2 6052 | . . . . 5 ⊢ ◡◡𝑆 = (𝑆 ↾ V) | |
5 | 4 | fveq1i 6673 | . . . 4 ⊢ (◡◡𝑆‘(𝐸‘ndx)) = ((𝑆 ↾ V)‘(𝐸‘ndx)) |
6 | fvex 6685 | . . . . 5 ⊢ (𝐸‘ndx) ∈ V | |
7 | fvres 6691 | . . . . 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 2846 | . . 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 3516 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ V) |
14 | opelxpi 5594 | . . . . . . 7 ⊢ (((𝐸‘ndx) ∈ V ∧ 𝐶 ∈ V) → 〈(𝐸‘ndx), 𝐶〉 ∈ (V × V)) | |
15 | 6, 13, 14 | sylancr 589 | . . . . . 6 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (V × V)) |
16 | 11, 15 | elind 4173 | . . . . 5 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (𝑆 ∩ (V × V))) |
17 | cnvcnv 6051 | . . . . 5 ⊢ ◡◡𝑆 = (𝑆 ∩ (V × V)) | |
18 | 16, 17 | eleqtrrdi 2926 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ ◡◡𝑆) |
19 | funopfv 6719 | . . . 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 2872 | . 2 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶) |
22 | 3, 21 | eqtr2d 2859 | 1 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∩ cin 3937 〈cop 4575 × cxp 5555 ◡ccnv 5556 ↾ cres 5559 Fun wfun 6351 ‘cfv 6357 ndxcnx 16482 Slot cslot 16484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 df-slot 16489 |
This theorem is referenced by: strfv2 16532 opelstrbas 16599 ebtwntg 26770 ecgrtg 26771 elntg 26772 edgfiedgval 26804 |
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