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Mirrors > Home > MPE Home > Th. List > funcsetcestrclem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for funcsetcestrc 18158. (Contributed by AV, 27-Mar-2020.) |
Ref | Expression |
---|---|
funcsetcestrc.s | ⊢ 𝑆 = (SetCat‘𝑈) |
funcsetcestrc.c | ⊢ 𝐶 = (Base‘𝑆) |
funcsetcestrc.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) |
funcsetcestrc.u | ⊢ (𝜑 → 𝑈 ∈ WUni) |
funcsetcestrc.o | ⊢ (𝜑 → ω ∈ 𝑈) |
funcsetcestrc.g | ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) |
Ref | Expression |
---|---|
funcsetcestrclem5 | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌 ↑m 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcsetcestrc.g | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) | |
2 | 1 | adantr 479 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) |
3 | oveq12 7428 | . . . . 5 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → (𝑦 ↑m 𝑥) = (𝑌 ↑m 𝑋)) | |
4 | 3 | ancoms 457 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑦 ↑m 𝑥) = (𝑌 ↑m 𝑋)) |
5 | 4 | reseq2d 5985 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ( I ↾ (𝑦 ↑m 𝑥)) = ( I ↾ (𝑌 ↑m 𝑋))) |
6 | 5 | adantl 480 | . 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ( I ↾ (𝑦 ↑m 𝑥)) = ( I ↾ (𝑌 ↑m 𝑋))) |
7 | simprl 769 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → 𝑋 ∈ 𝐶) | |
8 | simprr 771 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → 𝑌 ∈ 𝐶) | |
9 | ovexd 7454 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑌 ↑m 𝑋) ∈ V) | |
10 | 9 | resiexd 7228 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → ( I ↾ (𝑌 ↑m 𝑋)) ∈ V) |
11 | 2, 6, 7, 8, 10 | ovmpod 7573 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌 ↑m 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 {csn 4630 〈cop 4636 ↦ cmpt 5232 I cid 5575 ↾ cres 5680 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 ωcom 7871 ↑m cmap 8845 WUnicwun 10725 ndxcnx 17165 Basecbs 17183 SetCatcsetc 18067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 |
This theorem is referenced by: funcsetcestrclem6 18154 funcsetcestrclem7 18155 funcsetcestrclem8 18156 funcsetcestrclem9 18157 |
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