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Mirrors > Home > MPE Home > Th. List > pwssnf1o | Structured version Visualization version GIF version |
Description: Triviality of singleton powers: set equipollence. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
pwssnf1o.y | ⊢ 𝑌 = (𝑅 ↑s {𝐼}) |
pwssnf1o.b | ⊢ 𝐵 = (Base‘𝑅) |
pwssnf1o.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ({𝐼} × {𝑥})) |
pwssnf1o.c | ⊢ 𝐶 = (Base‘𝑌) |
Ref | Expression |
---|---|
pwssnf1o | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐵–1-1-onto→𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwssnf1o.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | 1 | fvexi 6423 | . . 3 ⊢ 𝐵 ∈ V |
3 | simpr 478 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐼 ∈ 𝑊) | |
4 | pwssnf1o.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ({𝐼} × {𝑥})) | |
5 | 4 | mapsnf1o 8187 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐵–1-1-onto→(𝐵 ↑𝑚 {𝐼})) |
6 | 2, 3, 5 | sylancr 582 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐵–1-1-onto→(𝐵 ↑𝑚 {𝐼})) |
7 | snex 5097 | . . . . . 6 ⊢ {𝐼} ∈ V | |
8 | pwssnf1o.y | . . . . . . 7 ⊢ 𝑌 = (𝑅 ↑s {𝐼}) | |
9 | 8, 1 | pwsbas 16459 | . . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ {𝐼} ∈ V) → (𝐵 ↑𝑚 {𝐼}) = (Base‘𝑌)) |
10 | 7, 9 | mpan2 683 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝐵 ↑𝑚 {𝐼}) = (Base‘𝑌)) |
11 | 10 | adantr 473 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝐵 ↑𝑚 {𝐼}) = (Base‘𝑌)) |
12 | pwssnf1o.c | . . . 4 ⊢ 𝐶 = (Base‘𝑌) | |
13 | 11, 12 | syl6reqr 2850 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐶 = (𝐵 ↑𝑚 {𝐼})) |
14 | f1oeq3 6345 | . . 3 ⊢ (𝐶 = (𝐵 ↑𝑚 {𝐼}) → (𝐹:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→(𝐵 ↑𝑚 {𝐼}))) | |
15 | 13, 14 | syl 17 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝐹:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→(𝐵 ↑𝑚 {𝐼}))) |
16 | 6, 15 | mpbird 249 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐵–1-1-onto→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3383 {csn 4366 ↦ cmpt 4920 × cxp 5308 –1-1-onto→wf1o 6098 ‘cfv 6099 (class class class)co 6876 ↑𝑚 cmap 8093 Basecbs 16181 ↑s cpws 16419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-map 8095 df-ixp 8147 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-sup 8588 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-fz 12577 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-plusg 16277 df-mulr 16278 df-sca 16280 df-vsca 16281 df-ip 16282 df-tset 16283 df-ple 16284 df-ds 16286 df-hom 16288 df-cco 16289 df-prds 16420 df-pws 16422 |
This theorem is referenced by: pwslnmlem1 38435 |
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