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Mirrors > Home > MPE Home > Th. List > funcsetcestrclem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for funcsetcestrc 17243. (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 |
---|---|
funcsetcestrclem4 | ⊢ (𝜑 → 𝐺 Fn (𝐶 × 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2795 | . . 3 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑𝑚 𝑥))) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑𝑚 𝑥))) | |
2 | ovex 7048 | . . . 4 ⊢ (𝑦 ↑𝑚 𝑥) ∈ V | |
3 | resiexg 7475 | . . . 4 ⊢ ((𝑦 ↑𝑚 𝑥) ∈ V → ( I ↾ (𝑦 ↑𝑚 𝑥)) ∈ V) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ( I ↾ (𝑦 ↑𝑚 𝑥)) ∈ V |
5 | 1, 4 | fnmpoi 7624 | . 2 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑𝑚 𝑥))) Fn (𝐶 × 𝐶) |
6 | funcsetcestrc.g | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑𝑚 𝑥)))) | |
7 | 6 | fneq1d 6316 | . 2 ⊢ (𝜑 → (𝐺 Fn (𝐶 × 𝐶) ↔ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑𝑚 𝑥))) Fn (𝐶 × 𝐶))) |
8 | 5, 7 | mpbiri 259 | 1 ⊢ (𝜑 → 𝐺 Fn (𝐶 × 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 Vcvv 3437 {csn 4472 〈cop 4478 ↦ cmpt 5041 I cid 5347 × cxp 5441 ↾ cres 5445 Fn wfn 6220 ‘cfv 6225 (class class class)co 7016 ∈ cmpo 7018 ωcom 7436 ↑𝑚 cmap 8256 WUnicwun 9968 ndxcnx 16309 Basecbs 16312 SetCatcsetc 17164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-1st 7545 df-2nd 7546 |
This theorem is referenced by: funcsetcestrc 17243 |
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