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Theorem imasdsf1o 24400

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 2735	. 2 ⊢ (ℝ^_𝑠 ↾_s (ℝ^ ∖ {-∞})) = (ℝ^_𝑠 ↾_s (ℝ^ ∖ {-∞}))
11		eqid 2735	. 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 2735	. 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 24399	1 ⊢ (𝜑 → ((𝐹‘𝑋)𝐷(𝐹‘𝑌)) = (𝑋𝐸𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ∖ cdif 3960 {csn 4631 ∪ ciun 4996 ↦ cmpt 5231 × cxp 5687 ran crn 5690 ↾ cres 5691 ∘ ccom 5693 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 1^st c1st 8011 2^nd c2nd 8012 ↑_m cmap 8865 1c1 11154 + caddc 11156 -∞cmnf 11291 ℝ^*cxr 11292 − cmin 11490 ℕcn 12264 ...cfz 13544 Basecbs 17245 ↾_s cress 17274 distcds 17307 Σ_g cgsu 17487 ℝ^*_𝑠cxrs 17547 “_s cimas 17551 ∞Metcxmet 21367

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-0g 17488 df-gsum 17489 df-xrs 17549 df-imas 17555 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-xmet 21375

This theorem is referenced by: imasf1oxmet 24401 imasf1omet 24402 xpsdsval 24407 imasf1obl 24517

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