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| Description: Lemma 4 for funcestrcsetc 18195. (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 2736 | . . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) | |
| 2 | ovex 7465 | . . . 4 ⊢ ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V | |
| 3 | resiexg 7935 | . . . 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 8096 | . 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 6660 | . 2 ⊢ (𝜑 → (𝐺 Fn (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) Fn (𝐵 × 𝐵))) | 
| 8 | 5, 7 | mpbiri 258 | 1 ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ↦ cmpt 5224 I cid 5576 × cxp 5682 ↾ cres 5686 Fn wfn 6555 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 ↑m cmap 8867 WUnicwun 10741 Basecbs 17248 SetCatcsetc 18121 ExtStrCatcestrc 18167 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 | 
| This theorem is referenced by: funcestrcsetc 18195 | 
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