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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 6117 | . . . 4 ⊢ (𝐴 ∈ dom 𝐺 → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐹‘𝐴)) | |
2 | 1 | eqcomd 2616 | . . 3 ⊢ (𝐴 ∈ dom 𝐺 → (𝐹‘𝐴) = ((𝐹 ↾ dom 𝐺)‘𝐴)) |
3 | funssres 5844 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺) | |
4 | 3 | fveq1d 6105 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐺‘𝐴)) |
5 | 2, 4 | sylan9eqr 2666 | . 2 ⊢ (((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴)) |
6 | 5 | 3impa 1251 | 1 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 dom cdm 5038 ↾ cres 5040 Fun wfun 5798 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: funsssuppss 7208 wfrlem12 7313 wfrlem14 7315 tfrlem9 7368 tfrlem11 7371 ac6sfi 8089 axdc3lem2 9156 axdc3lem4 9158 imasvscaval 16021 pserdv 23987 eupap1 26503 sspn 26975 bnj945 30098 bnj1502 30172 bnj545 30219 bnj548 30221 subfacp1lem2a 30416 subfacp1lem2b 30417 subfacp1lem5 30420 cvmliftlem10 30530 cvmliftlem13 30532 frrlem11 31036 subgruhgredgd 40508 subumgredg2 40509 subupgr 40511 |
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