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Theorem imasdsf1o 23527

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 2738	. 2 ⊢ (ℝ^_𝑠 ↾_s (ℝ^ ∖ {-∞})) = (ℝ^_𝑠 ↾_s (ℝ^ ∖ {-∞}))
11		eqid 2738	. 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 2738	. 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 23526	1 ⊢ (𝜑 → ((𝐹‘𝑋)𝐷(𝐹‘𝑌)) = (𝑋𝐸𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 ∖ cdif 3884 {csn 4561 ∪ ciun 4924 ↦ cmpt 5157 × cxp 5587 ran crn 5590 ↾ cres 5591 ∘ ccom 5593 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 ↑_m cmap 8615 1c1 10872 + caddc 10874 -∞cmnf 11007 ℝ^*cxr 11008 − cmin 11205 ℕcn 11973 ...cfz 13239 Basecbs 16912 ↾_s cress 16941 distcds 16971 Σ_g cgsu 17151 ℝ^*_𝑠cxrs 17211 “_s cimas 17215 ∞Metcxmet 20582

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-0g 17152 df-gsum 17153 df-xrs 17213 df-imas 17219 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-xmet 20590

This theorem is referenced by: imasf1oxmet 23528 imasf1omet 23529 xpsdsval 23534 imasf1obl 23644

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