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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnghmresfn | Structured version Visualization version GIF version |
Description: The class of non-unital ring homomorphisms restricted to subsets of non-unital rings is a function. (Contributed by AV, 4-Mar-2020.) |
Ref | Expression |
---|---|
rnghmresfn.b | ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) |
rnghmresfn.h | ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) |
Ref | Expression |
---|---|
rnghmresfn | ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnghmfn 44168 | . . 3 ⊢ RngHomo Fn (Rng × Rng) | |
2 | rnghmresfn.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) | |
3 | inss2 4208 | . . . . 5 ⊢ (𝑈 ∩ Rng) ⊆ Rng | |
4 | 2, 3 | eqsstrdi 4023 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ Rng) |
5 | xpss12 5572 | . . . 4 ⊢ ((𝐵 ⊆ Rng ∧ 𝐵 ⊆ Rng) → (𝐵 × 𝐵) ⊆ (Rng × Rng)) | |
6 | 4, 4, 5 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ (Rng × Rng)) |
7 | fnssres 6472 | . . 3 ⊢ (( RngHomo Fn (Rng × Rng) ∧ (𝐵 × 𝐵) ⊆ (Rng × Rng)) → ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) | |
8 | 1, 6, 7 | sylancr 589 | . 2 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) |
9 | rnghmresfn.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) | |
10 | 9 | fneq1d 6448 | . 2 ⊢ (𝜑 → (𝐻 Fn (𝐵 × 𝐵) ↔ ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))) |
11 | 8, 10 | mpbird 259 | 1 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3937 ⊆ wss 3938 × cxp 5555 ↾ cres 5559 Fn wfn 6352 Rngcrng 44152 RngHomo crngh 44163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-rnghomo 44165 |
This theorem is referenced by: rngcbas 44243 rngchomfval 44244 rngchomfeqhom 44247 rngccofval 44248 dfrngc2 44250 rnghmsubcsetc 44255 rngcid 44257 funcrngcsetc 44276 |
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