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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 20382 | . . 3 ⊢ RngHom Fn (Rng × Rng) | |
2 | rnghmresfn.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) | |
3 | inss2 4224 | . . . . 5 ⊢ (𝑈 ∩ Rng) ⊆ Rng | |
4 | 2, 3 | eqsstrdi 4027 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ Rng) |
5 | xpss12 5687 | . . . 4 ⊢ ((𝐵 ⊆ Rng ∧ 𝐵 ⊆ Rng) → (𝐵 × 𝐵) ⊆ (Rng × Rng)) | |
6 | 4, 4, 5 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ (Rng × Rng)) |
7 | fnssres 6673 | . . 3 ⊢ (( RngHom Fn (Rng × Rng) ∧ (𝐵 × 𝐵) ⊆ (Rng × Rng)) → ( RngHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) | |
8 | 1, 6, 7 | sylancr 585 | . 2 ⊢ (𝜑 → ( RngHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) |
9 | rnghmresfn.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵))) | |
10 | 9 | fneq1d 6642 | . 2 ⊢ (𝜑 → (𝐻 Fn (𝐵 × 𝐵) ↔ ( RngHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))) |
11 | 8, 10 | mpbird 256 | 1 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∩ cin 3938 ⊆ wss 3939 × cxp 5670 ↾ cres 5674 Fn wfn 6538 Rngcrng 20096 RngHom crnghm 20377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-rnghm 20379 |
This theorem is referenced by: rngcbas 20558 rngchomfval 20559 rngchomfeqhom 20562 rngccofval 20563 dfrngc2 20565 rnghmsubcsetc 20570 rngcid 20572 funcrngcsetc 20577 |
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