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Mirrors > Home > MPE Home > Th. List > tcphex | Structured version Visualization version GIF version |
Description: Lemma for tcphbas 25265 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 2734 | . . 3 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) | |
2 | fvrn0 6949 | . . . 4 ⊢ (√‘(𝑥 , 𝑥)) ∈ (ran √ ∪ {∅}) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝑉 → (√‘(𝑥 , 𝑥)) ∈ (ran √ ∪ {∅})) |
4 | 1, 3 | fmpti 7144 | . 2 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))):𝑉⟶(ran √ ∪ {∅}) |
5 | tcphex.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
6 | 5 | fvexi 6933 | . 2 ⊢ 𝑉 ∈ V |
7 | cnex 11261 | . . . 4 ⊢ ℂ ∈ V | |
8 | sqrtf 15408 | . . . . 5 ⊢ √:ℂ⟶ℂ | |
9 | frn 6753 | . . . . 5 ⊢ (√:ℂ⟶ℂ → ran √ ⊆ ℂ) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ran √ ⊆ ℂ |
11 | 7, 10 | ssexi 5343 | . . 3 ⊢ ran √ ∈ V |
12 | p0ex 5405 | . . 3 ⊢ {∅} ∈ V | |
13 | 11, 12 | unex 7775 | . 2 ⊢ (ran √ ∪ {∅}) ∈ V |
14 | fex2 7970 | . 2 ⊢ (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))):𝑉⟶(ran √ ∪ {∅}) ∧ 𝑉 ∈ V ∧ (ran √ ∪ {∅}) ∈ V) → (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) ∈ V) | |
15 | 4, 6, 13, 14 | mp3an 1461 | 1 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2103 Vcvv 3482 ∪ cun 3968 ⊆ wss 3970 ∅c0 4347 {csn 4648 ↦ cmpt 5252 ran crn 5700 ⟶wf 6568 ‘cfv 6572 (class class class)co 7445 ℂcc 11178 √csqrt 15278 Basecbs 17253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-sup 9507 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-n0 12550 df-z 12636 df-uz 12900 df-rp 13054 df-seq 14049 df-exp 14109 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 |
This theorem is referenced by: tcphbas 25265 tchplusg 25266 tcphmulr 25268 tcphsca 25269 tcphvsca 25270 tcphip 25271 tcphtopn 25272 tcphds 25277 rrxdim 33619 |
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