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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 9670 | . . 3 ⊢ inr:V–1-1-onto→({1o} × V) | |
2 | f1ofun 6715 | . . 3 ⊢ (inr:V–1-1-onto→({1o} × V) → Fun inr) | |
3 | ffvresb 6993 | . . 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 3449 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ V) | |
6 | opex 5382 | . . . . 5 ⊢ 〈1o, 𝑥〉 ∈ V | |
7 | df-inr 9660 | . . . . 5 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
8 | 6, 7 | dmmpti 6574 | . . . 4 ⊢ dom inr = V |
9 | 5, 8 | eleqtrrdi 2852 | . . 3 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ dom inr) |
10 | djurcl 9668 | . . 3 ⊢ (𝑥 ∈ 𝐵 → (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵)) | |
11 | 9, 10 | jca 512 | . 2 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴 ⊔ 𝐵))) |
12 | 4, 11 | mprgbir 3081 | 1 ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∈ wcel 2110 ∀wral 3066 Vcvv 3431 {csn 4567 〈cop 4573 × cxp 5587 dom cdm 5589 ↾ cres 5591 Fun wfun 6425 ⟶wf 6427 –1-1-onto→wf1o 6430 ‘cfv 6431 1oc1o 8279 ⊔ cdju 9655 inrcinr 9657 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7580 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-om 7705 df-1st 7822 df-2nd 7823 df-1o 8286 df-dju 9658 df-inr 9660 |
This theorem is referenced by: inrresf1 9674 updjudhcoinrg 9690 updjud 9691 |
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