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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 6925 | . . . 4 ⊢ (𝐴 ∈ dom 𝐺 → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐹‘𝐴)) | |
| 2 | 1 | eqcomd 2743 | . . 3 ⊢ (𝐴 ∈ dom 𝐺 → (𝐹‘𝐴) = ((𝐹 ↾ dom 𝐺)‘𝐴)) |
| 3 | funssres 6610 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺) | |
| 4 | 3 | fveq1d 6908 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐺‘𝐴)) |
| 5 | 2, 4 | sylan9eqr 2799 | . 2 ⊢ (((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴)) |
| 6 | 5 | 3impa 1110 | 1 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 dom cdm 5685 ↾ cres 5687 Fun wfun 6555 ‘cfv 6561 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: fviunfun 7969 funelss 8072 funsssuppss 8215 frrlem10 8320 wfrlem12OLD 8360 wfrlem14OLD 8362 tfrlem9 8425 tfrlem11 8428 ac6sfi 9320 axdc3lem2 10491 axdc3lem4 10493 imasvscaval 17583 pserdv 26473 subgruhgredgd 29301 subumgredg2 29302 subupgr 29304 sspn 30755 bnj945 34787 bnj1502 34862 bnj545 34909 bnj548 34911 subfacp1lem2a 35185 subfacp1lem2b 35186 subfacp1lem5 35189 cvmliftlem10 35299 cvmliftlem13 35301 |
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