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Mirrors > Home > ILE Home > Th. List > fliftel1 | GIF version |
Description: Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
flift.1 | ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) |
flift.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) |
flift.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) |
Ref | Expression |
---|---|
fliftel1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴𝐹𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flift.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) | |
2 | flift.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) | |
3 | opexg 4200 | . . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 〈𝐴, 𝐵〉 ∈ V) | |
4 | 1, 2, 3 | syl2anc 409 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ V) |
5 | eqid 2164 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
6 | 5 | elrnmpt1 4849 | . . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 〈𝐴, 𝐵〉 ∈ V) → 〈𝐴, 𝐵〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
7 | 6 | adantll 468 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 〈𝐴, 𝐵〉 ∈ V) → 〈𝐴, 𝐵〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
8 | 4, 7 | mpdan 418 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
9 | flift.1 | . . 3 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
10 | 8, 9 | eleqtrrdi 2258 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ 𝐹) |
11 | df-br 3977 | . 2 ⊢ (𝐴𝐹𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹) | |
12 | 10, 11 | sylibr 133 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴𝐹𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 Vcvv 2721 〈cop 3573 class class class wbr 3976 ↦ cmpt 4037 ran crn 4599 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-mpt 4039 df-cnv 4606 df-dm 4608 df-rn 4609 |
This theorem is referenced by: fliftfun 5758 qliftel1 6573 |
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