Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > djuf1olemr | GIF version |
Description: Lemma for djulf1or 7045 and djurf1or 7046. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 7042. (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 4896 | . . 3 ⊢ (𝐹 ↾ 𝐴) = ((𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩) ↾ 𝐴) |
4 | ssv 3175 | . . . 4 ⊢ 𝐴 ⊆ V | |
5 | resmpt 4948 | . . . 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 2196 | . 2 ⊢ (𝐹 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ ⟨𝑋, 𝑥⟩) |
8 | 1, 7 | djuf1olem 7042 | 1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→({𝑋} × 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2146 Vcvv 2735 ⊆ wss 3127 {csn 3589 ⟨cop 3592 ↦ cmpt 4059 × cxp 4618 ↾ cres 4622 –1-1-onto→wf1o 5207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-sbc 2961 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-1st 6131 df-2nd 6132 |
This theorem is referenced by: djulf1or 7045 djurf1or 7046 |
Copyright terms: Public domain | W3C validator |