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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 4150 | . . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 〈𝐴, 𝐵〉 ∈ V) | |
4 | 1, 2, 3 | syl2anc 408 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ V) |
5 | eqid 2139 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
6 | 5 | elrnmpt1 4790 | . . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 〈𝐴, 𝐵〉 ∈ V) → 〈𝐴, 𝐵〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
7 | 6 | adantll 467 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 〈𝐴, 𝐵〉 ∈ V) → 〈𝐴, 𝐵〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
8 | 4, 7 | mpdan 417 | . . 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 2233 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ 𝐹) |
11 | df-br 3930 | . 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 1331 ∈ wcel 1480 Vcvv 2686 〈cop 3530 class class class wbr 3929 ↦ cmpt 3989 ran crn 4540 |
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-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-mpt 3991 df-cnv 4547 df-dm 4549 df-rn 4550 |
This theorem is referenced by: fliftfun 5697 qliftel1 6510 |
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