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Mirrors > Home > MPE Home > Th. List > fliftel1 | Structured version Visualization version 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 | opex 5321 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
2 | eqid 2798 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
3 | 2 | elrnmpt1 5794 | . . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 〈𝐴, 𝐵〉 ∈ V) → 〈𝐴, 𝐵〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
4 | 1, 3 | mpan2 690 | . . . 4 ⊢ (𝑥 ∈ 𝑋 → 〈𝐴, 𝐵〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
5 | 4 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
6 | flift.1 | . . 3 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
7 | 5, 6 | eleqtrrdi 2901 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ 𝐹) |
8 | df-br 5031 | . 2 ⊢ (𝐴𝐹𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹) | |
9 | 7, 8 | sylibr 237 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴𝐹𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rex 3112 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-mpt 5111 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: fliftfun 7044 qliftel1 8364 |
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