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Mirrors > Home > MPE Home > Th. List > Mathboxes > clrellem | Structured version Visualization version GIF version |
Description: When the property 𝜓 holds for a relation substituted for 𝑥, then the closure on that property is a relation if the base set is a relation. (Contributed by RP, 30-Jul-2020.) |
Ref | Expression |
---|---|
clrellem.y | ⊢ (𝜑 → 𝑌 ∈ V) |
clrellem.rel | ⊢ (𝜑 → Rel 𝑋) |
clrellem.sub | ⊢ (𝑥 = ◡◡𝑌 → (𝜓 ↔ 𝜒)) |
clrellem.sup | ⊢ (𝜑 → 𝑋 ⊆ 𝑌) |
clrellem.maj | ⊢ (𝜑 → 𝜒) |
Ref | Expression |
---|---|
clrellem | ⊢ (𝜑 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clrellem.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ V) | |
2 | cnvexg 7648 | . . . 4 ⊢ (𝑌 ∈ V → ◡𝑌 ∈ V) | |
3 | cnvexg 7648 | . . . 4 ⊢ (◡𝑌 ∈ V → ◡◡𝑌 ∈ V) | |
4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (𝜑 → ◡◡𝑌 ∈ V) |
5 | clrellem.rel | . . . . . 6 ⊢ (𝜑 → Rel 𝑋) | |
6 | dfrel2 6015 | . . . . . 6 ⊢ (Rel 𝑋 ↔ ◡◡𝑋 = 𝑋) | |
7 | 5, 6 | sylib 221 | . . . . 5 ⊢ (𝜑 → ◡◡𝑋 = 𝑋) |
8 | clrellem.sup | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑌) | |
9 | cnvss 5709 | . . . . . 6 ⊢ (𝑋 ⊆ 𝑌 → ◡𝑋 ⊆ ◡𝑌) | |
10 | cnvss 5709 | . . . . . 6 ⊢ (◡𝑋 ⊆ ◡𝑌 → ◡◡𝑋 ⊆ ◡◡𝑌) | |
11 | 8, 9, 10 | 3syl 18 | . . . . 5 ⊢ (𝜑 → ◡◡𝑋 ⊆ ◡◡𝑌) |
12 | 7, 11 | eqsstrrd 3914 | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ ◡◡𝑌) |
13 | clrellem.maj | . . . 4 ⊢ (𝜑 → 𝜒) | |
14 | relcnv 5935 | . . . . 5 ⊢ Rel ◡◡𝑌 | |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → Rel ◡◡𝑌) |
16 | 12, 13, 15 | jca31 518 | . . 3 ⊢ (𝜑 → ((𝑋 ⊆ ◡◡𝑌 ∧ 𝜒) ∧ Rel ◡◡𝑌)) |
17 | clrellem.sub | . . . . 5 ⊢ (𝑥 = ◡◡𝑌 → (𝜓 ↔ 𝜒)) | |
18 | 17 | cleq2lem 40745 | . . . 4 ⊢ (𝑥 = ◡◡𝑌 → ((𝑋 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑋 ⊆ ◡◡𝑌 ∧ 𝜒))) |
19 | releq 5616 | . . . 4 ⊢ (𝑥 = ◡◡𝑌 → (Rel 𝑥 ↔ Rel ◡◡𝑌)) | |
20 | 18, 19 | anbi12d 634 | . . 3 ⊢ (𝑥 = ◡◡𝑌 → (((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥) ↔ ((𝑋 ⊆ ◡◡𝑌 ∧ 𝜒) ∧ Rel ◡◡𝑌))) |
21 | 4, 16, 20 | spcedv 3500 | . 2 ⊢ (𝜑 → ∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥)) |
22 | releq 5616 | . . . 4 ⊢ (𝑦 = 𝑥 → (Rel 𝑦 ↔ Rel 𝑥)) | |
23 | 22 | rexab2 3597 | . . 3 ⊢ (∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦 ↔ ∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥)) |
24 | 23 | biimpri 231 | . 2 ⊢ (∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥) → ∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦) |
25 | relint 5657 | . 2 ⊢ (∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}) | |
26 | 21, 24, 25 | 3syl 18 | 1 ⊢ (𝜑 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2113 {cab 2716 ∃wrex 3054 Vcvv 3397 ⊆ wss 3841 ∩ cint 4833 ◡ccnv 5518 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-int 4834 df-iin 4881 df-br 5028 df-opab 5090 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: (None) |
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