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Mirrors > Home > MPE Home > Th. List > inrresf | Structured version Visualization version GIF version |
Description: The right injection restricted to the right class of a disjoint union is a function from the right class into the disjoint union. (Contributed by AV, 27-Jun-2022.) |
Ref | Expression |
---|---|
inrresf | ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djurf1o 9341 | . . 3 ⊢ inr:V–1-1-onto→({1o} × V) | |
2 | f1ofun 6616 | . . 3 ⊢ (inr:V–1-1-onto→({1o} × V) → Fun inr) | |
3 | ffvresb 6887 | . . 3 ⊢ (Fun inr → ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵)))) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵))) |
5 | elex 3512 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ V) | |
6 | opex 5355 | . . . . 5 ⊢ 〈1o, 𝑥〉 ∈ V | |
7 | df-inr 9331 | . . . . 5 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
8 | 6, 7 | dmmpti 6491 | . . . 4 ⊢ dom inr = V |
9 | 5, 8 | eleqtrrdi 2924 | . . 3 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ dom inr) |
10 | djurcl 9339 | . . 3 ⊢ (𝑥 ∈ 𝐵 → (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵)) | |
11 | 9, 10 | jca 514 | . 2 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵))) |
12 | 4, 11 | mprgbir 3153 | 1 ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 {csn 4566 〈cop 4572 × cxp 5552 dom cdm 5554 ↾ cres 5556 Fun wfun 6348 ⟶wf 6350 –1-1-onto→wf1o 6353 ‘cfv 6354 1oc1o 8094 ⊔ cdju 9326 inrcinr 9328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-om 7580 df-1st 7688 df-2nd 7689 df-1o 8101 df-dju 9329 df-inr 9331 |
This theorem is referenced by: inrresf1 9345 updjudhcoinrg 9361 updjud 9362 |
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