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Mirrors > Home > ILE Home > Th. List > djuf1olemr | GIF version |
Description: Lemma for djulf1or 6941 and djurf1or 6942. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 6938. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.) |
Ref | Expression |
---|---|
djuf1olemr.1 | ⊢ 𝑋 ∈ V |
djuf1olemr.2 | ⊢ 𝐹 = (𝑥 ∈ V ↦ 〈𝑋, 𝑥〉) |
Ref | Expression |
---|---|
djuf1olemr | ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→({𝑋} × 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djuf1olemr.1 | . 2 ⊢ 𝑋 ∈ V | |
2 | djuf1olemr.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ V ↦ 〈𝑋, 𝑥〉) | |
3 | 2 | reseq1i 4815 | . . 3 ⊢ (𝐹 ↾ 𝐴) = ((𝑥 ∈ V ↦ 〈𝑋, 𝑥〉) ↾ 𝐴) |
4 | ssv 3119 | . . . 4 ⊢ 𝐴 ⊆ V | |
5 | resmpt 4867 | . . . 4 ⊢ (𝐴 ⊆ V → ((𝑥 ∈ V ↦ 〈𝑋, 𝑥〉) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 〈𝑋, 𝑥〉)) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ ((𝑥 ∈ V ↦ 〈𝑋, 𝑥〉) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 〈𝑋, 𝑥〉) |
7 | 3, 6 | eqtri 2160 | . 2 ⊢ (𝐹 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 〈𝑋, 𝑥〉) |
8 | 1, 7 | djuf1olem 6938 | 1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→({𝑋} × 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 {csn 3527 〈cop 3530 ↦ cmpt 3989 × cxp 4537 ↾ cres 4541 –1-1-onto→wf1o 5122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: djulf1or 6941 djurf1or 6942 |
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