Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > embedsetcestrclem | Structured version Visualization version GIF version |
Description: Lemma for embedsetcestrc 17419. (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 17408 | . 2 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) |
9 | 1, 2, 3 | funcsetcestrclem1 17406 | . . . . . 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 17406 | . . . . . 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 2839 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ {〈(Base‘ndx), 𝑦〉} = {〈(Base‘ndx), 𝑧〉})) |
14 | opex 5358 | . . . . . 6 ⊢ 〈(Base‘ndx), 𝑦〉 ∈ V | |
15 | sneqbg 4776 | . . . . . 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 6687 | . . . . . . 7 ⊢ (𝜑 → (Base‘ndx) ∈ V) | |
18 | simpl 485 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ 𝐶) | |
19 | opthg 5371 | . . . . . . 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 3193 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) |
26 | dff13 7015 | . 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 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 {csn 4569 〈cop 4575 ↦ cmpt 5148 ⟶wf 6353 –1-1→wf1 6354 ‘cfv 6357 ωcom 7582 WUnicwun 10124 ndxcnx 16482 Basecbs 16485 SetCatcsetc 17337 ExtStrCatcestrc 17374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-wun 10126 df-ni 10296 df-pli 10297 df-mi 10298 df-lti 10299 df-plpq 10332 df-mpq 10333 df-ltpq 10334 df-enq 10335 df-nq 10336 df-erq 10337 df-plq 10338 df-mq 10339 df-1nq 10340 df-rq 10341 df-ltnq 10342 df-np 10405 df-plp 10407 df-ltp 10409 df-enr 10479 df-nr 10480 df-c 10545 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-hom 16591 df-cco 16592 df-setc 17338 df-estrc 17375 |
This theorem is referenced by: embedsetcestrc 17419 |
Copyright terms: Public domain | W3C validator |