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Mirrors > Home > ILE Home > Th. List > Mathboxes > djucllem | GIF version |
Description: Lemma for djulcl 7044 and djurcl 7045. (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 5535 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴)) | |
2 | elex 2748 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | djucllem.1 | . . . . . 6 ⊢ 𝑋 ∈ V | |
4 | 3 | snid 3622 | . . . . 5 ⊢ 𝑋 ∈ {𝑋} |
5 | opelxpi 4655 | . . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝐴 ∈ 𝐵) → 〈𝑋, 𝐴〉 ∈ ({𝑋} × 𝐵)) | |
6 | 4, 5 | mpan 424 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 〈𝑋, 𝐴〉 ∈ ({𝑋} × 𝐵)) |
7 | opeq2 3777 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈𝑋, 𝑥〉 = 〈𝑋, 𝐴〉) | |
8 | djucllem.2 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ V ↦ 〈𝑋, 𝑥〉) | |
9 | 7, 8 | fvmptg 5588 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 〈𝑋, 𝐴〉 ∈ ({𝑋} × 𝐵)) → (𝐹‘𝐴) = 〈𝑋, 𝐴〉) |
10 | 2, 6, 9 | syl2anc 411 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐹‘𝐴) = 〈𝑋, 𝐴〉) |
11 | 1, 10 | eqtrd 2210 | . 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = 〈𝑋, 𝐴〉) |
12 | 11, 6 | eqeltrd 2254 | 1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 {csn 3591 〈cop 3594 ↦ cmpt 4061 × cxp 4621 ↾ cres 4625 ‘cfv 5212 |
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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-res 4635 df-iota 5174 df-fun 5214 df-fv 5220 |
This theorem is referenced by: djulclALT 14202 djurclALT 14203 |
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