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Mirrors > Home > MPE Home > Th. List > funcsetcestrclem6 | Structured version Visualization version GIF version |
Description: Lemma 6 for funcsetcestrc 16627. (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 |
---|---|
funcsetcestrclem6 | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑𝑚 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcsetcestrc.s | . . . . 5 ⊢ 𝑆 = (SetCat‘𝑈) | |
2 | funcsetcestrc.c | . . . . 5 ⊢ 𝐶 = (Base‘𝑆) | |
3 | funcsetcestrc.f | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) | |
4 | funcsetcestrc.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
5 | funcsetcestrc.o | . . . . 5 ⊢ (𝜑 → ω ∈ 𝑈) | |
6 | funcsetcestrc.g | . . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑𝑚 𝑥)))) | |
7 | 1, 2, 3, 4, 5, 6 | funcsetcestrclem5 16622 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌 ↑𝑚 𝑋))) |
8 | 7 | 3adant3 1074 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑𝑚 𝑋)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌 ↑𝑚 𝑋))) |
9 | 8 | fveq1d 6105 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑𝑚 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = (( I ↾ (𝑌 ↑𝑚 𝑋))‘𝐻)) |
10 | fvresi 6344 | . . 3 ⊢ (𝐻 ∈ (𝑌 ↑𝑚 𝑋) → (( I ↾ (𝑌 ↑𝑚 𝑋))‘𝐻) = 𝐻) | |
11 | 10 | 3ad2ant3 1077 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑𝑚 𝑋)) → (( I ↾ (𝑌 ↑𝑚 𝑋))‘𝐻) = 𝐻) |
12 | 9, 11 | eqtrd 2644 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑𝑚 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {csn 4125 〈cop 4131 ↦ cmpt 4643 I cid 4948 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 ↑𝑚 cmap 7744 WUnicwun 9401 ndxcnx 15692 Basecbs 15695 SetCatcsetc 16548 |
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-rep 4699 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-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-iun 4457 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-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: funcsetcestrclem9 16626 fthsetcestrc 16628 fullsetcestrc 16629 |
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