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Mirrors > Home > MPE Home > Th. List > setsidvald | Structured version Visualization version GIF version |
Description: Value of the structure replacement function, deduction version. (Contributed by AV, 14-Mar-2020.) |
Ref | Expression |
---|---|
setsidvald.e | ⊢ 𝐸 = Slot 𝑁 |
setsidvald.n | ⊢ 𝑁 ∈ ℕ |
setsidvald.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
setsidvald.f | ⊢ (𝜑 → Fun 𝑆) |
setsidvald.d | ⊢ (𝜑 → (𝐸‘ndx) ∈ dom 𝑆) |
Ref | Expression |
---|---|
setsidvald | ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsidvald.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
2 | fvex 6686 | . . 3 ⊢ (𝐸‘𝑆) ∈ V | |
3 | setsval 16516 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐸‘𝑆) ∈ V) → (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉})) | |
4 | 1, 2, 3 | sylancl 588 | . 2 ⊢ (𝜑 → (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉})) |
5 | setsidvald.e | . . . . . . 7 ⊢ 𝐸 = Slot 𝑁 | |
6 | setsidvald.n | . . . . . . 7 ⊢ 𝑁 ∈ ℕ | |
7 | 5, 6 | ndxid 16512 | . . . . . 6 ⊢ 𝐸 = Slot (𝐸‘ndx) |
8 | 7, 1 | strfvnd 16505 | . . . . 5 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
9 | 8 | opeq2d 4813 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), (𝐸‘𝑆)〉 = 〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉) |
10 | 9 | sneqd 4582 | . . 3 ⊢ (𝜑 → {〈(𝐸‘ndx), (𝐸‘𝑆)〉} = {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) |
11 | 10 | uneq2d 4142 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉}) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉})) |
12 | setsidvald.f | . . 3 ⊢ (𝜑 → Fun 𝑆) | |
13 | setsidvald.d | . . 3 ⊢ (𝜑 → (𝐸‘ndx) ∈ dom 𝑆) | |
14 | funresdfunsn 6954 | . . 3 ⊢ ((Fun 𝑆 ∧ (𝐸‘ndx) ∈ dom 𝑆) → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) = 𝑆) | |
15 | 12, 13, 14 | syl2anc 586 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) = 𝑆) |
16 | 4, 11, 15 | 3eqtrrd 2864 | 1 ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∖ cdif 3936 ∪ cun 3937 {csn 4570 〈cop 4576 dom cdm 5558 ↾ cres 5560 Fun wfun 6352 ‘cfv 6358 (class class class)co 7159 ℕcn 11641 ndxcnx 16483 sSet csts 16484 Slot cslot 16485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-1cn 10598 ax-addcl 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-nn 11642 df-ndx 16489 df-slot 16490 df-sets 16493 |
This theorem is referenced by: ressval3d 16564 |
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