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Mirrors > Home > ILE Home > Th. List > Mathboxes > djucllem | GIF version |
Description: Lemma for djulcl 6944 and djurcl 6945. (Contributed by BJ, 4-Jul-2022.) |
Ref | Expression |
---|---|
djucllem.1 | ⊢ 𝑋 ∈ V |
djucllem.2 | ⊢ 𝐹 = (𝑥 ∈ V ↦ 〈𝑋, 𝑥〉) |
Ref | Expression |
---|---|
djucllem | ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 5453 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴)) | |
2 | elex 2700 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | djucllem.1 | . . . . . 6 ⊢ 𝑋 ∈ V | |
4 | 3 | snid 3563 | . . . . 5 ⊢ 𝑋 ∈ {𝑋} |
5 | opelxpi 4579 | . . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝐴 ∈ 𝐵) → 〈𝑋, 𝐴〉 ∈ ({𝑋} × 𝐵)) | |
6 | 4, 5 | mpan 421 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 〈𝑋, 𝐴〉 ∈ ({𝑋} × 𝐵)) |
7 | opeq2 3714 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈𝑋, 𝑥〉 = 〈𝑋, 𝐴〉) | |
8 | djucllem.2 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ V ↦ 〈𝑋, 𝑥〉) | |
9 | 7, 8 | fvmptg 5505 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 〈𝑋, 𝐴〉 ∈ ({𝑋} × 𝐵)) → (𝐹‘𝐴) = 〈𝑋, 𝐴〉) |
10 | 2, 6, 9 | syl2anc 409 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐹‘𝐴) = 〈𝑋, 𝐴〉) |
11 | 1, 10 | eqtrd 2173 | . 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = 〈𝑋, 𝐴〉) |
12 | 11, 6 | eqeltrd 2217 | 1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 Vcvv 2689 {csn 3532 〈cop 3535 ↦ cmpt 3997 × cxp 4545 ↾ cres 4549 ‘cfv 5131 |
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-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 |
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-res 4559 df-iota 5096 df-fun 5133 df-fv 5139 |
This theorem is referenced by: djulclALT 13179 djurclALT 13180 |
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