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Mirrors > Home > MPE Home > Th. List > embedsetcestrclem | Structured version Visualization version GIF version |
Description: Lemma for embedsetcestrc 17411. (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 17400 | . 2 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) |
9 | 1, 2, 3 | funcsetcestrclem1 17398 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝐹‘𝑦) = {〈(Base‘ndx), 𝑦〉}) |
10 | 9 | adantrr 715 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝐹‘𝑦) = {〈(Base‘ndx), 𝑦〉}) |
11 | 1, 2, 3 | funcsetcestrclem1 17398 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) = {〈(Base‘ndx), 𝑧〉}) |
12 | 11 | adantrl 714 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝐹‘𝑧) = {〈(Base‘ndx), 𝑧〉}) |
13 | 10, 12 | eqeq12d 2837 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ {〈(Base‘ndx), 𝑦〉} = {〈(Base‘ndx), 𝑧〉})) |
14 | opex 5348 | . . . . . 6 ⊢ 〈(Base‘ndx), 𝑦〉 ∈ V | |
15 | sneqbg 4767 | . . . . . 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 6679 | . . . . . . 7 ⊢ (𝜑 → (Base‘ndx) ∈ V) | |
18 | simpl 485 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ 𝐶) | |
19 | opthg 5361 | . . . . . . 7 ⊢ (((Base‘ndx) ∈ V ∧ 𝑦 ∈ 𝐶) → (〈(Base‘ndx), 𝑦〉 = 〈(Base‘ndx), 𝑧〉 ↔ ((Base‘ndx) = (Base‘ndx) ∧ 𝑦 = 𝑧))) | |
20 | 17, 18, 19 | syl2an 597 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (〈(Base‘ndx), 𝑦〉 = 〈(Base‘ndx), 𝑧〉 ↔ ((Base‘ndx) = (Base‘ndx) ∧ 𝑦 = 𝑧))) |
21 | simpr 487 | . . . . . 6 ⊢ (((Base‘ndx) = (Base‘ndx) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧) | |
22 | 20, 21 | syl6bi 255 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (〈(Base‘ndx), 𝑦〉 = 〈(Base‘ndx), 𝑧〉 → 𝑦 = 𝑧)) |
23 | 16, 22 | sylbid 242 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ({〈(Base‘ndx), 𝑦〉} = {〈(Base‘ndx), 𝑧〉} → 𝑦 = 𝑧)) |
24 | 13, 23 | sylbid 242 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) |
25 | 24 | ralrimivva 3191 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) |
26 | dff13 7007 | . 2 ⊢ (𝐹:𝐶–1-1→𝐵 ↔ (𝐹:𝐶⟶𝐵 ∧ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))) | |
27 | 8, 25, 26 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐹:𝐶–1-1→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 {csn 4560 〈cop 4566 ↦ cmpt 5138 ⟶wf 6345 –1-1→wf1 6346 ‘cfv 6349 ωcom 7574 WUnicwun 10116 ndxcnx 16474 Basecbs 16477 SetCatcsetc 17329 ExtStrCatcestrc 17366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-wun 10118 df-ni 10288 df-pli 10289 df-mi 10290 df-lti 10291 df-plpq 10324 df-mpq 10325 df-ltpq 10326 df-enq 10327 df-nq 10328 df-erq 10329 df-plq 10330 df-mq 10331 df-1nq 10332 df-rq 10333 df-ltnq 10334 df-np 10397 df-plp 10399 df-ltp 10401 df-enr 10471 df-nr 10472 df-c 10537 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-hom 16583 df-cco 16584 df-setc 17330 df-estrc 17367 |
This theorem is referenced by: embedsetcestrc 17411 |
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