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Mirrors > Home > MPE Home > Th. List > tcphex | Structured version Visualization version GIF version |
Description: Lemma for tcphbas 23386 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
tcphex.v | ⊢ 𝑉 = (Base‘𝑊) |
Ref | Expression |
---|---|
tcphex | ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2824 | . . 3 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) | |
2 | fvrn0 6460 | . . . 4 ⊢ (√‘(𝑥 , 𝑥)) ∈ (ran √ ∪ {∅}) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝑉 → (√‘(𝑥 , 𝑥)) ∈ (ran √ ∪ {∅})) |
4 | 1, 3 | fmpti 6630 | . 2 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))):𝑉⟶(ran √ ∪ {∅}) |
5 | tcphex.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
6 | 5 | fvexi 6446 | . 2 ⊢ 𝑉 ∈ V |
7 | cnex 10332 | . . . 4 ⊢ ℂ ∈ V | |
8 | sqrtf 14479 | . . . . 5 ⊢ √:ℂ⟶ℂ | |
9 | frn 6283 | . . . . 5 ⊢ (√:ℂ⟶ℂ → ran √ ⊆ ℂ) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ran √ ⊆ ℂ |
11 | 7, 10 | ssexi 5027 | . . 3 ⊢ ran √ ∈ V |
12 | p0ex 5082 | . . 3 ⊢ {∅} ∈ V | |
13 | 11, 12 | unex 7215 | . 2 ⊢ (ran √ ∪ {∅}) ∈ V |
14 | fex2 7382 | . 2 ⊢ (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))):𝑉⟶(ran √ ∪ {∅}) ∧ 𝑉 ∈ V ∧ (ran √ ∪ {∅}) ∈ V) → (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) ∈ V) | |
15 | 4, 6, 13, 14 | mp3an 1591 | 1 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∈ wcel 2166 Vcvv 3413 ∪ cun 3795 ⊆ wss 3797 ∅c0 4143 {csn 4396 ↦ cmpt 4951 ran crn 5342 ⟶wf 6118 ‘cfv 6122 (class class class)co 6904 ℂcc 10249 √csqrt 14349 Basecbs 16221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-sup 8616 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-n0 11618 df-z 11704 df-uz 11968 df-rp 12112 df-seq 13095 df-exp 13154 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 |
This theorem is referenced by: tcphbas 23386 tchplusg 23387 tcphmulr 23389 tcphsca 23390 tcphvsca 23391 tcphip 23392 tcphtopn 23393 tcphds 23398 |
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