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Mirrors > Home > ILE Home > Th. List > djuf1olemr | GIF version |
Description: Lemma for djulf1or 6934 and djurf1or 6935. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 6931. (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 4810 | . . 3 ⊢ (𝐹 ↾ 𝐴) = ((𝑥 ∈ V ↦ 〈𝑋, 𝑥〉) ↾ 𝐴) |
4 | ssv 3114 | . . . 4 ⊢ 𝐴 ⊆ V | |
5 | resmpt 4862 | . . . 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 2158 | . 2 ⊢ (𝐹 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 〈𝑋, 𝑥〉) |
8 | 1, 7 | djuf1olem 6931 | 1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→({𝑋} × 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2681 ⊆ wss 3066 {csn 3522 〈cop 3525 ↦ cmpt 3984 × cxp 4532 ↾ cres 4536 –1-1-onto→wf1o 5117 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-1st 6031 df-2nd 6032 |
This theorem is referenced by: djulf1or 6934 djurf1or 6935 |
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