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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 9134 | . . 3 ⊢ inr:V–1-1-onto→({1o} × V) | |
2 | f1ofun 6443 | . . 3 ⊢ (inr:V–1-1-onto→({1o} × V) → Fun inr) | |
3 | ffvresb 6709 | . . 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 3426 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ V) | |
6 | opex 5209 | . . . . 5 ⊢ 〈1o, 𝑥〉 ∈ V | |
7 | df-inr 9124 | . . . . 5 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
8 | 6, 7 | dmmpti 6319 | . . . 4 ⊢ dom inr = V |
9 | 5, 8 | syl6eleqr 2870 | . . 3 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ dom inr) |
10 | djurcl 9132 | . . 3 ⊢ (𝑥 ∈ 𝐵 → (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵)) | |
11 | 9, 10 | jca 504 | . 2 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵))) |
12 | 4, 11 | mprgbir 3096 | 1 ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 ∈ wcel 2051 ∀wral 3081 Vcvv 3408 {csn 4435 〈cop 4441 × cxp 5401 dom cdm 5403 ↾ cres 5405 Fun wfun 6179 ⟶wf 6181 –1-1-onto→wf1o 6184 ‘cfv 6185 1oc1o 7896 ⊔ cdju 9119 inrcinr 9121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-om 7395 df-1st 7499 df-2nd 7500 df-1o 7903 df-dju 9122 df-inr 9124 |
This theorem is referenced by: inrresf1 9138 updjudhcoinrg 9154 updjud 9155 |
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