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Theorem imasdsf1o 24384

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 2737	. 2 ⊢ (ℝ^_𝑠 ↾_s (ℝ^ ∖ {-∞})) = (ℝ^_𝑠 ↾_s (ℝ^ ∖ {-∞}))
11		eqid 2737	. 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 2737	. 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 24383	1 ⊢ (𝜑 → ((𝐹‘𝑋)𝐷(𝐹‘𝑌)) = (𝑋𝐸𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ∖ cdif 3948 {csn 4626 ∪ ciun 4991 ↦ cmpt 5225 × cxp 5683 ran crn 5686 ↾ cres 5687 ∘ ccom 5689 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 1^st c1st 8012 2^nd c2nd 8013 ↑_m cmap 8866 1c1 11156 + caddc 11158 -∞cmnf 11293 ℝ^*cxr 11294 − cmin 11492 ℕcn 12266 ...cfz 13547 Basecbs 17247 ↾_s cress 17274 distcds 17306 Σ_g cgsu 17485 ℝ^*_𝑠cxrs 17545 “_s cimas 17549 ∞Metcxmet 21349

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-gsum 17487 df-xrs 17547 df-imas 17553 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-xmet 21357

This theorem is referenced by: imasf1oxmet 24385 imasf1omet 24386 xpsdsval 24391 imasf1obl 24501

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