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| Mirrors > Home > MPE Home > Th. List > embedsetcestrclem | Structured version Visualization version GIF version | ||
| Description: Lemma for embedsetcestrc 18212. (Contributed by AV, 31-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) |
| funcsetcestrclem3.e | ⊢ 𝐸 = (ExtStrCat‘𝑈) |
| funcsetcestrclem3.b | ⊢ 𝐵 = (Base‘𝐸) |
| Ref | Expression |
|---|---|
| embedsetcestrclem | ⊢ (𝜑 → 𝐹:𝐶–1-1→𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funcsetcestrc.s | . . 3 ⊢ 𝑆 = (SetCat‘𝑈) | |
| 2 | funcsetcestrc.c | . . 3 ⊢ 𝐶 = (Base‘𝑆) | |
| 3 | funcsetcestrc.f | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) | |
| 4 | funcsetcestrc.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
| 5 | funcsetcestrc.o | . . 3 ⊢ (𝜑 → ω ∈ 𝑈) | |
| 6 | funcsetcestrclem3.e | . . 3 ⊢ 𝐸 = (ExtStrCat‘𝑈) | |
| 7 | funcsetcestrclem3.b | . . 3 ⊢ 𝐵 = (Base‘𝐸) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | funcsetcestrclem3 18201 | . 2 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) |
| 9 | 1, 2, 3 | funcsetcestrclem1 18199 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝐹‘𝑦) = {〈(Base‘ndx), 𝑦〉}) |
| 10 | 9 | adantrr 717 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝐹‘𝑦) = {〈(Base‘ndx), 𝑦〉}) |
| 11 | 1, 2, 3 | funcsetcestrclem1 18199 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) = {〈(Base‘ndx), 𝑧〉}) |
| 12 | 11 | adantrl 716 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝐹‘𝑧) = {〈(Base‘ndx), 𝑧〉}) |
| 13 | 10, 12 | eqeq12d 2753 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ {〈(Base‘ndx), 𝑦〉} = {〈(Base‘ndx), 𝑧〉})) |
| 14 | opex 5469 | . . . . . 6 ⊢ 〈(Base‘ndx), 𝑦〉 ∈ V | |
| 15 | sneqbg 4843 | . . . . . 6 ⊢ (〈(Base‘ndx), 𝑦〉 ∈ V → ({〈(Base‘ndx), 𝑦〉} = {〈(Base‘ndx), 𝑧〉} ↔ 〈(Base‘ndx), 𝑦〉 = 〈(Base‘ndx), 𝑧〉)) | |
| 16 | 14, 15 | mp1i 13 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ({〈(Base‘ndx), 𝑦〉} = {〈(Base‘ndx), 𝑧〉} ↔ 〈(Base‘ndx), 𝑦〉 = 〈(Base‘ndx), 𝑧〉)) |
| 17 | fvexd 6921 | . . . . . . 7 ⊢ (𝜑 → (Base‘ndx) ∈ V) | |
| 18 | simpl 482 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ 𝐶) | |
| 19 | opthg 5482 | . . . . . . 7 ⊢ (((Base‘ndx) ∈ V ∧ 𝑦 ∈ 𝐶) → (〈(Base‘ndx), 𝑦〉 = 〈(Base‘ndx), 𝑧〉 ↔ ((Base‘ndx) = (Base‘ndx) ∧ 𝑦 = 𝑧))) | |
| 20 | 17, 18, 19 | syl2an 596 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (〈(Base‘ndx), 𝑦〉 = 〈(Base‘ndx), 𝑧〉 ↔ ((Base‘ndx) = (Base‘ndx) ∧ 𝑦 = 𝑧))) |
| 21 | simpr 484 | . . . . . 6 ⊢ (((Base‘ndx) = (Base‘ndx) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧) | |
| 22 | 20, 21 | biimtrdi 253 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (〈(Base‘ndx), 𝑦〉 = 〈(Base‘ndx), 𝑧〉 → 𝑦 = 𝑧)) |
| 23 | 16, 22 | sylbid 240 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ({〈(Base‘ndx), 𝑦〉} = {〈(Base‘ndx), 𝑧〉} → 𝑦 = 𝑧)) |
| 24 | 13, 23 | sylbid 240 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) |
| 25 | 24 | ralrimivva 3202 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) |
| 26 | dff13 7275 | . 2 ⊢ (𝐹:𝐶–1-1→𝐵 ↔ (𝐹:𝐶⟶𝐵 ∧ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))) | |
| 27 | 8, 25, 26 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐹:𝐶–1-1→𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 {csn 4626 〈cop 4632 ↦ cmpt 5225 ⟶wf 6557 –1-1→wf1 6558 ‘cfv 6561 ωcom 7887 WUnicwun 10740 ndxcnx 17230 Basecbs 17247 SetCatcsetc 18120 ExtStrCatcestrc 18166 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 df-er 8745 df-ec 8747 df-qs 8751 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-wun 10742 df-ni 10912 df-pli 10913 df-mi 10914 df-lti 10915 df-plpq 10948 df-mpq 10949 df-ltpq 10950 df-enq 10951 df-nq 10952 df-erq 10953 df-plq 10954 df-mq 10955 df-1nq 10956 df-rq 10957 df-ltnq 10958 df-np 11021 df-plp 11023 df-ltp 11025 df-enr 11095 df-nr 11096 df-c 11161 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-hom 17321 df-cco 17322 df-setc 18121 df-estrc 18167 |
| This theorem is referenced by: embedsetcestrc 18212 |
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