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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 46278 | . . 3 ⊢ RngHomo Fn (Rng × Rng) | |
2 | rnghmresfn.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) | |
3 | inss2 4193 | . . . . 5 ⊢ (𝑈 ∩ Rng) ⊆ Rng | |
4 | 2, 3 | eqsstrdi 4002 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ Rng) |
5 | xpss12 5652 | . . . 4 ⊢ ((𝐵 ⊆ Rng ∧ 𝐵 ⊆ Rng) → (𝐵 × 𝐵) ⊆ (Rng × Rng)) | |
6 | 4, 4, 5 | syl2anc 585 | . . 3 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ (Rng × Rng)) |
7 | fnssres 6628 | . . 3 ⊢ (( RngHomo Fn (Rng × Rng) ∧ (𝐵 × 𝐵) ⊆ (Rng × Rng)) → ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) | |
8 | 1, 6, 7 | sylancr 588 | . 2 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) |
9 | rnghmresfn.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) | |
10 | 9 | fneq1d 6599 | . 2 ⊢ (𝜑 → (𝐻 Fn (𝐵 × 𝐵) ↔ ( RngHomo ↾ (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 1542 ∩ cin 3913 ⊆ wss 3914 × cxp 5635 ↾ cres 5639 Fn wfn 6495 Rngcrng 46262 RngHomo crngh 46273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-rnghomo 46275 |
This theorem is referenced by: rngcbas 46353 rngchomfval 46354 rngchomfeqhom 46357 rngccofval 46358 dfrngc2 46360 rnghmsubcsetc 46365 rngcid 46367 funcrngcsetc 46386 |
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