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Mirrors > Home > MPE Home > Th. List > funcestrcsetclem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for funcestrcsetc 17401. (Contributed by AV, 23-Mar-2020.) |
Ref | Expression |
---|---|
funcestrcsetc.e | ⊢ 𝐸 = (ExtStrCat‘𝑈) |
funcestrcsetc.s | ⊢ 𝑆 = (SetCat‘𝑈) |
funcestrcsetc.b | ⊢ 𝐵 = (Base‘𝐸) |
funcestrcsetc.c | ⊢ 𝐶 = (Base‘𝑆) |
funcestrcsetc.u | ⊢ (𝜑 → 𝑈 ∈ WUni) |
funcestrcsetc.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) |
funcestrcsetc.g | ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) |
funcestrcsetc.m | ⊢ 𝑀 = (Base‘𝑋) |
funcestrcsetc.n | ⊢ 𝑁 = (Base‘𝑌) |
Ref | Expression |
---|---|
funcestrcsetclem5 | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑m 𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcestrcsetc.g | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) | |
2 | 1 | adantr 483 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) |
3 | fveq2 6672 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (Base‘𝑦) = (Base‘𝑌)) | |
4 | fveq2 6672 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋)) | |
5 | 3, 4 | oveqan12rd 7178 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑌) ↑m (Base‘𝑋))) |
6 | funcestrcsetc.n | . . . . . 6 ⊢ 𝑁 = (Base‘𝑌) | |
7 | funcestrcsetc.m | . . . . . 6 ⊢ 𝑀 = (Base‘𝑋) | |
8 | 6, 7 | oveq12i 7170 | . . . . 5 ⊢ (𝑁 ↑m 𝑀) = ((Base‘𝑌) ↑m (Base‘𝑋)) |
9 | 5, 8 | syl6eqr 2876 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((Base‘𝑦) ↑m (Base‘𝑥)) = (𝑁 ↑m 𝑀)) |
10 | 9 | reseq2d 5855 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) = ( I ↾ (𝑁 ↑m 𝑀))) |
11 | 10 | adantl 484 | . 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) = ( I ↾ (𝑁 ↑m 𝑀))) |
12 | simprl 769 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
13 | simprr 771 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
14 | ovexd 7193 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑁 ↑m 𝑀) ∈ V) | |
15 | 14 | resiexd 6981 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ (𝑁 ↑m 𝑀)) ∈ V) |
16 | 2, 11, 12, 13, 15 | ovmpod 7304 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑m 𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ↦ cmpt 5148 I cid 5461 ↾ cres 5559 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 ↑m cmap 8408 WUnicwun 10124 Basecbs 16485 SetCatcsetc 17337 ExtStrCatcestrc 17374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 |
This theorem is referenced by: funcestrcsetclem6 17397 funcestrcsetclem7 17398 funcestrcsetclem8 17399 funcestrcsetclem9 17400 |
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