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Mirrors > Home > ILE Home > Th. List > djuf1olemr | GIF version |
Description: Lemma for djulf1or 6949 and djurf1or 6950. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 6946. (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 4823 | . . 3 ⊢ (𝐹 ↾ 𝐴) = ((𝑥 ∈ V ↦ 〈𝑋, 𝑥〉) ↾ 𝐴) |
4 | ssv 3124 | . . . 4 ⊢ 𝐴 ⊆ V | |
5 | resmpt 4875 | . . . 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 2161 | . 2 ⊢ (𝐹 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 〈𝑋, 𝑥〉) |
8 | 1, 7 | djuf1olem 6946 | 1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→({𝑋} × 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 {csn 3532 〈cop 3535 ↦ cmpt 3997 × cxp 4545 ↾ cres 4549 –1-1-onto→wf1o 5130 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-1st 6046 df-2nd 6047 |
This theorem is referenced by: djulf1or 6949 djurf1or 6950 |
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