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Mirrors > Home > ILE Home > Th. List > Mathboxes > djucllem | GIF version |
Description: Lemma for djulcl 6936 and djurcl 6937. (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 5445 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴)) | |
2 | elex 2697 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | djucllem.1 | . . . . . 6 ⊢ 𝑋 ∈ V | |
4 | 3 | snid 3556 | . . . . 5 ⊢ 𝑋 ∈ {𝑋} |
5 | opelxpi 4571 | . . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝐴 ∈ 𝐵) → 〈𝑋, 𝐴〉 ∈ ({𝑋} × 𝐵)) | |
6 | 4, 5 | mpan 420 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 〈𝑋, 𝐴〉 ∈ ({𝑋} × 𝐵)) |
7 | opeq2 3706 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈𝑋, 𝑥〉 = 〈𝑋, 𝐴〉) | |
8 | djucllem.2 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ V ↦ 〈𝑋, 𝑥〉) | |
9 | 7, 8 | fvmptg 5497 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 〈𝑋, 𝐴〉 ∈ ({𝑋} × 𝐵)) → (𝐹‘𝐴) = 〈𝑋, 𝐴〉) |
10 | 2, 6, 9 | syl2anc 408 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐹‘𝐴) = 〈𝑋, 𝐴〉) |
11 | 1, 10 | eqtrd 2172 | . 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = 〈𝑋, 𝐴〉) |
12 | 11, 6 | eqeltrd 2216 | 1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 {csn 3527 〈cop 3530 ↦ cmpt 3989 × cxp 4537 ↾ cres 4541 ‘cfv 5123 |
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-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 |
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-res 4551 df-iota 5088 df-fun 5125 df-fv 5131 |
This theorem is referenced by: djulclALT 13008 djurclALT 13009 |
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