![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > embedsetcestrclem | Structured version Visualization version GIF version |
Description: Lemma for embedsetcestrc 18121. (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 18110 | . 2 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) |
9 | 1, 2, 3 | funcsetcestrclem1 18108 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝐹‘𝑦) = {〈(Base‘ndx), 𝑦〉}) |
10 | 9 | adantrr 714 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝐹‘𝑦) = {〈(Base‘ndx), 𝑦〉}) |
11 | 1, 2, 3 | funcsetcestrclem1 18108 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) = {〈(Base‘ndx), 𝑧〉}) |
12 | 11 | adantrl 713 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝐹‘𝑧) = {〈(Base‘ndx), 𝑧〉}) |
13 | 10, 12 | eqeq12d 2740 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ {〈(Base‘ndx), 𝑦〉} = {〈(Base‘ndx), 𝑧〉})) |
14 | opex 5454 | . . . . . 6 ⊢ 〈(Base‘ndx), 𝑦〉 ∈ V | |
15 | sneqbg 4836 | . . . . . 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 6896 | . . . . . . 7 ⊢ (𝜑 → (Base‘ndx) ∈ V) | |
18 | simpl 482 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ 𝐶) | |
19 | opthg 5467 | . . . . . . 7 ⊢ (((Base‘ndx) ∈ V ∧ 𝑦 ∈ 𝐶) → (〈(Base‘ndx), 𝑦〉 = 〈(Base‘ndx), 𝑧〉 ↔ ((Base‘ndx) = (Base‘ndx) ∧ 𝑦 = 𝑧))) | |
20 | 17, 18, 19 | syl2an 595 | . . . . . 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 | syl6bi 253 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (〈(Base‘ndx), 𝑦〉 = 〈(Base‘ndx), 𝑧〉 → 𝑦 = 𝑧)) |
23 | 16, 22 | sylbid 239 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ({〈(Base‘ndx), 𝑦〉} = {〈(Base‘ndx), 𝑧〉} → 𝑦 = 𝑧)) |
24 | 13, 23 | sylbid 239 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) |
25 | 24 | ralrimivva 3192 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) |
26 | dff13 7246 | . 2 ⊢ (𝐹:𝐶–1-1→𝐵 ↔ (𝐹:𝐶⟶𝐵 ∧ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))) | |
27 | 8, 25, 26 | sylanbrc 582 | 1 ⊢ (𝜑 → 𝐹:𝐶–1-1→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 {csn 4620 〈cop 4626 ↦ cmpt 5221 ⟶wf 6529 –1-1→wf1 6530 ‘cfv 6533 ωcom 7848 WUnicwun 10691 ndxcnx 17125 Basecbs 17143 SetCatcsetc 18027 ExtStrCatcestrc 18075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-omul 8466 df-er 8699 df-ec 8701 df-qs 8705 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-wun 10693 df-ni 10863 df-pli 10864 df-mi 10865 df-lti 10866 df-plpq 10899 df-mpq 10900 df-ltpq 10901 df-enq 10902 df-nq 10903 df-erq 10904 df-plq 10905 df-mq 10906 df-1nq 10907 df-rq 10908 df-ltnq 10909 df-np 10972 df-plp 10974 df-ltp 10976 df-enr 11046 df-nr 11047 df-c 11112 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17079 df-slot 17114 df-ndx 17126 df-base 17144 df-hom 17220 df-cco 17221 df-setc 18028 df-estrc 18076 |
This theorem is referenced by: embedsetcestrc 18121 |
Copyright terms: Public domain | W3C validator |