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Mirrors > Home > MPE Home > Th. List > funssfv | Structured version Visualization version GIF version |
Description: The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.) |
Ref | Expression |
---|---|
funssfv | ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 6465 | . . . 4 ⊢ (𝐴 ∈ dom 𝐺 → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐹‘𝐴)) | |
2 | 1 | eqcomd 2783 | . . 3 ⊢ (𝐴 ∈ dom 𝐺 → (𝐹‘𝐴) = ((𝐹 ↾ dom 𝐺)‘𝐴)) |
3 | funssres 6178 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺) | |
4 | 3 | fveq1d 6448 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐺‘𝐴)) |
5 | 2, 4 | sylan9eqr 2835 | . 2 ⊢ (((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴)) |
6 | 5 | 3impa 1097 | 1 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ⊆ wss 3791 dom cdm 5355 ↾ cres 5357 Fun wfun 6129 ‘cfv 6135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-res 5367 df-iota 6099 df-fun 6137 df-fv 6143 |
This theorem is referenced by: funsssuppss 7603 wfrlem12 7709 wfrlem14 7711 tfrlem9 7764 tfrlem11 7767 ac6sfi 8492 axdc3lem2 9608 axdc3lem4 9610 imasvscaval 16584 pserdv 24620 subgruhgredgd 26631 subumgredg2 26632 subupgr 26634 sspn 28163 bnj945 31443 bnj1502 31517 bnj545 31564 bnj548 31566 subfacp1lem2a 31761 subfacp1lem2b 31762 subfacp1lem5 31765 cvmliftlem10 31875 cvmliftlem13 31877 frrlem11 32381 |
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