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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 4230 | . . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ⟨𝐴, 𝐵⟩ ∈ V) | |
4 | 1, 2, 3 | syl2anc 411 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ V) |
5 | eqid 2177 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) | |
6 | 5 | elrnmpt1 4880 | . . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)) |
7 | 6 | adantll 476 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)) |
8 | 4, 7 | mpdan 421 | . . 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 2271 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ 𝐹) |
11 | df-br 4006 | . 2 ⊢ (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹) | |
12 | 10, 11 | sylibr 134 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴𝐹𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2739 ⟨cop 3597 class class class wbr 4005 ↦ cmpt 4066 ran crn 4629 |
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 4123 ax-pow 4176 ax-pr 4211 |
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-rex 2461 df-v 2741 df-sbc 2965 df-csb 3060 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-br 4006 df-opab 4067 df-mpt 4068 df-cnv 4636 df-dm 4638 df-rn 4639 |
This theorem is referenced by: fliftfun 5799 qliftel1 6618 |
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