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Mirrors > Home > MPE Home > Th. List > csbfv | Structured version Visualization version GIF version |
Description: Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.) |
Ref | Expression |
---|---|
csbfv | ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbfv2g 6373 | . . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘⦋𝐴 / 𝑥⦌𝑥)) | |
2 | csbvarg 4147 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) | |
3 | 2 | fveq2d 6336 | . . 3 ⊢ (𝐴 ∈ V → (𝐹‘⦋𝐴 / 𝑥⦌𝑥) = (𝐹‘𝐴)) |
4 | 1, 3 | eqtrd 2805 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)) |
5 | csbprc 4124 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = ∅) | |
6 | fvprc 6326 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
7 | 5, 6 | eqtr4d 2808 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)) |
8 | 4, 7 | pm2.61i 176 | 1 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ⦋csb 3682 ∅c0 4063 ‘cfv 6031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-nul 4923 ax-pow 4974 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-dm 5259 df-iota 5994 df-fv 6039 |
This theorem is referenced by: mptcoe1fsupp 19800 mptcoe1matfsupp 20827 mp2pm2mplem4 20834 chfacfscmulfsupp 20884 chfacfpmmulfsupp 20888 cpmidpmatlem3 20897 cayhamlem4 20913 cayleyhamilton1 20917 logbmpt 24747 nbgrcl 26450 nbgrclOLD 26451 nbgrnvtx0 26455 iuninc 29717 disjxpin 29739 cnfinltrel 33578 finixpnum 33727 cdlemkid3N 36742 cdlemkid4 36743 cdlemk39s 36748 mccllem 40347 |
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