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Mirrors > Home > MPE Home > Th. List > funcestrcsetclem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for funcestrcsetc 17232. (Contributed by AV, 22-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))))) |
Ref | Expression |
---|---|
funcestrcsetclem4 | ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2797 | . . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) | |
2 | ovex 7055 | . . . 4 ⊢ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V | |
3 | resiexg 7482 | . . . 4 ⊢ (((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V → ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V |
5 | 1, 4 | fnmpoi 7631 | . 2 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) Fn (𝐵 × 𝐵) |
6 | funcestrcsetc.g | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))) | |
7 | 6 | fneq1d 6323 | . 2 ⊢ (𝜑 → (𝐺 Fn (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) Fn (𝐵 × 𝐵))) |
8 | 5, 7 | mpbiri 259 | 1 ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1525 ∈ wcel 2083 Vcvv 3440 ↦ cmpt 5047 I cid 5354 × cxp 5448 ↾ cres 5452 Fn wfn 6227 ‘cfv 6232 (class class class)co 7023 ∈ cmpo 7025 ↑𝑚 cmap 8263 WUnicwun 9975 Basecbs 16316 SetCatcsetc 17168 ExtStrCatcestrc 17205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-fv 6240 df-ov 7026 df-oprab 7027 df-mpo 7028 df-1st 7552 df-2nd 7553 |
This theorem is referenced by: funcestrcsetc 17232 |
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