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Theorem imasdsf1o 24405

Description: The distance function is transferred across an image structure under a bijection. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Hypotheses**
Ref	Expression
imasdsf1o.u	⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
imasdsf1o.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasdsf1o.f	⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
imasdsf1o.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imasdsf1o.e	⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
imasdsf1o.d	⊢ 𝐷 = (dist‘𝑈)
imasdsf1o.m	⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))
imasdsf1o.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
imasdsf1o.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)

**Assertion**
Ref	Expression
imasdsf1o	⊢ (𝜑 → ((𝐹‘𝑋)𝐷(𝐹‘𝑌)) = (𝑋𝐸𝑌))

Proof of Theorem imasdsf1o Dummy variables 𝑔 ℎ 𝑖 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasdsf1o.u	. 2 ⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
2		imasdsf1o.v	. 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasdsf1o.f	. 2 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
4		imasdsf1o.r	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
5		imasdsf1o.e	. 2 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
6		imasdsf1o.d	. 2 ⊢ 𝐷 = (dist‘𝑈)
7		imasdsf1o.m	. 2 ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))
8		imasdsf1o.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
9		imasdsf1o.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
10		eqid 2740	. 2 ⊢ (ℝ^_𝑠 ↾_s (ℝ^ ∖ {-∞})) = (ℝ^_𝑠 ↾_s (ℝ^ ∖ {-∞}))
11		eqid 2740	. 2 ⊢ {ℎ ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)) ∣ ((𝐹‘(1^st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2^nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2^nd ‘(ℎ‘𝑖))) = (𝐹‘(1^st ‘(ℎ‘(𝑖 + 1)))))} = {ℎ ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)) ∣ ((𝐹‘(1^st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2^nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2^nd ‘(ℎ‘𝑖))) = (𝐹‘(1^st ‘(ℎ‘(𝑖 + 1)))))}
12		eqid 2740	. 2 ⊢ ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)) ∣ ((𝐹‘(1^st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2^nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2^nd ‘(ℎ‘𝑖))) = (𝐹‘(1^st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))) = ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)) ∣ ((𝐹‘(1^st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2^nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2^nd ‘(ℎ‘𝑖))) = (𝐹‘(1^st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔)))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	imasdsf1olem 24404	1 ⊢ (𝜑 → ((𝐹‘𝑋)𝐷(𝐹‘𝑌)) = (𝑋𝐸𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 ∖ cdif 3973 {csn 4648 ∪ ciun 5015 ↦ cmpt 5249 × cxp 5698 ran crn 5701 ↾ cres 5702 ∘ ccom 5704 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 ↑_m cmap 8884 1c1 11185 + caddc 11187 -∞cmnf 11322 ℝ^*cxr 11323 − cmin 11520 ℕcn 12293 ...cfz 13567 Basecbs 17258 ↾_s cress 17287 distcds 17320 Σ_g cgsu 17500 ℝ^*_𝑠cxrs 17560 “_s cimas 17564 ∞Metcxmet 21372

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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-0g 17501 df-gsum 17502 df-xrs 17562 df-imas 17568 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-xmet 21380

This theorem is referenced by: imasf1oxmet 24406 imasf1omet 24407 xpsdsval 24412 imasf1obl 24522

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