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Mirrors > Home > MPE Home > Th. List > fliftrel | Structured version Visualization version GIF version |
Description: 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (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 |
---|---|
fliftrel | ⊢ (𝜑 → 𝐹 ⊆ (𝑅 × 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flift.1 | . 2 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
2 | flift.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) | |
3 | flift.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) | |
4 | opelxpi 5072 | . . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) | |
5 | 2, 3, 4 | syl2anc 691 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) |
6 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
7 | 5, 6 | fmptd 6292 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉):𝑋⟶(𝑅 × 𝑆)) |
8 | frn 5966 | . . 3 ⊢ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉):𝑋⟶(𝑅 × 𝑆) → ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) ⊆ (𝑅 × 𝑆)) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) ⊆ (𝑅 × 𝑆)) |
10 | 1, 9 | syl5eqss 3612 | 1 ⊢ (𝜑 → 𝐹 ⊆ (𝑅 × 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 〈cop 4131 ↦ cmpt 4643 × cxp 5036 ran crn 5039 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 |
This theorem is referenced by: fliftcnv 6461 fliftfun 6462 fliftf 6465 qliftrel 7716 fmucndlem 21905 pi1xfrcnv 22665 |
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