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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 20439 | . . 3 ⊢ RngHom Fn (Rng × Rng) | |
| 2 | rnghmresfn.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) | |
| 3 | inss2 4238 | . . . . 5 ⊢ (𝑈 ∩ Rng) ⊆ Rng | |
| 4 | 2, 3 | eqsstrdi 4028 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ Rng) | 
| 5 | xpss12 5700 | . . . 4 ⊢ ((𝐵 ⊆ Rng ∧ 𝐵 ⊆ Rng) → (𝐵 × 𝐵) ⊆ (Rng × Rng)) | |
| 6 | 4, 4, 5 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ (Rng × Rng)) | 
| 7 | fnssres 6691 | . . 3 ⊢ (( RngHom Fn (Rng × Rng) ∧ (𝐵 × 𝐵) ⊆ (Rng × Rng)) → ( RngHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) | |
| 8 | 1, 6, 7 | sylancr 587 | . 2 ⊢ (𝜑 → ( RngHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) | 
| 9 | rnghmresfn.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵))) | |
| 10 | 9 | fneq1d 6661 | . 2 ⊢ (𝜑 → (𝐻 Fn (𝐵 × 𝐵) ↔ ( RngHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))) | 
| 11 | 8, 10 | mpbird 257 | 1 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∩ cin 3950 ⊆ wss 3951 × cxp 5683 ↾ cres 5687 Fn wfn 6556 Rngcrng 20149 RngHom crnghm 20434 | 
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-rnghm 20436 | 
| This theorem is referenced by: rngcbas 20621 rngchomfval 20622 rngchomfeqhom 20625 rngccofval 20626 dfrngc2 20628 rnghmsubcsetc 20633 rngcid 20635 funcrngcsetc 20640 | 
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