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| Mirrors > Home > MPE Home > Th. List > 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 = ( RngHom ↾ (𝐵 × 𝐵))) |
| Ref | Expression |
|---|---|
| rnghmresfn | ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rnghmfn 20489 | . . 3 ⊢ RngHom Fn (Rng × Rng) | |
| 2 | rnghmresfn.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) | |
| 3 | inss2 4190 | . . . . 5 ⊢ (𝑈 ∩ Rng) ⊆ Rng | |
| 4 | 2, 3 | eqsstrdi 3981 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ Rng) |
| 5 | xpss12 5663 | . . . 4 ⊢ ((𝐵 ⊆ Rng ∧ 𝐵 ⊆ Rng) → (𝐵 × 𝐵) ⊆ (Rng × Rng)) | |
| 6 | 4, 4, 5 | syl2anc 593 | . . 3 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ (Rng × Rng)) |
| 7 | fnssres 6645 | . . 3 ⊢ (( RngHom Fn (Rng × Rng) ∧ (𝐵 × 𝐵) ⊆ (Rng × Rng)) → ( RngHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) | |
| 8 | 1, 6, 7 | sylancr 596 | . 2 ⊢ (𝜑 → ( RngHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) |
| 9 | rnghmresfn.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵))) | |
| 10 | 9 | fneq1d 6615 | . 2 ⊢ (𝜑 → (𝐻 Fn (𝐵 × 𝐵) ↔ ( RngHom ↾ (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 1561 ∩ cin 3904 ⊆ wss 3905 × cxp 5646 ↾ cres 5650 Fn wfn 6517 Rngcrng 20199 RngHom crnghm 20484 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5247 ax-nul 5257 ax-pr 5391 ax-un 7719 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3078 df-rex 3088 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5102 df-opab 5164 df-mpt 5183 df-id 5543 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-fv 6530 df-ov 7400 df-oprab 7401 df-mpo 7402 df-1st 7971 df-2nd 7972 df-rnghm 20486 |
| This theorem is referenced by: rngcbas 20672 rngchomfval 20673 rngchomfeqhom 20676 rngccofval 20677 dfrngc2 20679 rnghmsubcsetc 20684 rngcid 20686 funcrngcsetc 20691 |
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