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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mrsubcn | Structured version Visualization version GIF version |
Description: A substitution does not change the value of constant substrings. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mrsubccat.s | ⊢ 𝑆 = (mRSubst‘𝑇) |
mrsubccat.r | ⊢ 𝑅 = (mREx‘𝑇) |
mrsubcn.v | ⊢ 𝑉 = (mVR‘𝑇) |
mrsubcn.c | ⊢ 𝐶 = (mCN‘𝑇) |
Ref | Expression |
---|---|
mrsubcn | ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ (𝐶 ∖ 𝑉)) → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4346 | . . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅) | |
2 | mrsubccat.s | . . . . . 6 ⊢ 𝑆 = (mRSubst‘𝑇) | |
3 | 2 | rnfvprc 6901 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅) |
4 | 1, 3 | nsyl2 141 | . . . 4 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V) |
5 | mrsubcn.v | . . . . 5 ⊢ 𝑉 = (mVR‘𝑇) | |
6 | mrsubccat.r | . . . . 5 ⊢ 𝑅 = (mREx‘𝑇) | |
7 | 5, 6, 2 | mrsubff 35497 | . . . 4 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅)) |
8 | ffun 6740 | . . . 4 ⊢ (𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅) → Fun 𝑆) | |
9 | 4, 7, 8 | 3syl 18 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → Fun 𝑆) |
10 | 5, 6, 2 | mrsubrn 35498 | . . . . 5 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑m 𝑉)) |
11 | 10 | eleq2i 2831 | . . . 4 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ (𝑆 “ (𝑅 ↑m 𝑉))) |
12 | 11 | biimpi 216 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹 ∈ (𝑆 “ (𝑅 ↑m 𝑉))) |
13 | fvelima 6974 | . . 3 ⊢ ((Fun 𝑆 ∧ 𝐹 ∈ (𝑆 “ (𝑅 ↑m 𝑉))) → ∃𝑓 ∈ (𝑅 ↑m 𝑉)(𝑆‘𝑓) = 𝐹) | |
14 | 9, 12, 13 | syl2anc 584 | . 2 ⊢ (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ (𝑅 ↑m 𝑉)(𝑆‘𝑓) = 𝐹) |
15 | elmapi 8888 | . . . . . . 7 ⊢ (𝑓 ∈ (𝑅 ↑m 𝑉) → 𝑓:𝑉⟶𝑅) | |
16 | 15 | adantl 481 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → 𝑓:𝑉⟶𝑅) |
17 | ssidd 4019 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → 𝑉 ⊆ 𝑉) | |
18 | eldifi 4141 | . . . . . . . 8 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → 𝑋 ∈ 𝐶) | |
19 | elun1 4192 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝐶 ∪ 𝑉)) | |
20 | 18, 19 | syl 17 | . . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → 𝑋 ∈ (𝐶 ∪ 𝑉)) |
21 | 20 | adantr 480 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → 𝑋 ∈ (𝐶 ∪ 𝑉)) |
22 | mrsubcn.c | . . . . . . 7 ⊢ 𝐶 = (mCN‘𝑇) | |
23 | 22, 5, 6, 2 | mrsubcv 35495 | . . . . . 6 ⊢ ((𝑓:𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉)) |
24 | 16, 17, 21, 23 | syl3anc 1370 | . . . . 5 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉)) |
25 | eldifn 4142 | . . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → ¬ 𝑋 ∈ 𝑉) | |
26 | 25 | adantr 480 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ¬ 𝑋 ∈ 𝑉) |
27 | 26 | iffalsed 4542 | . . . . 5 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉) = 〈“𝑋”〉) |
28 | 24, 27 | eqtrd 2775 | . . . 4 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = 〈“𝑋”〉) |
29 | fveq1 6906 | . . . . 5 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘〈“𝑋”〉) = (𝐹‘〈“𝑋”〉)) | |
30 | 29 | eqeq1d 2737 | . . . 4 ⊢ ((𝑆‘𝑓) = 𝐹 → (((𝑆‘𝑓)‘〈“𝑋”〉) = 〈“𝑋”〉 ↔ (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
31 | 28, 30 | syl5ibcom 245 | . . 3 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ((𝑆‘𝑓) = 𝐹 → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
32 | 31 | rexlimdva 3153 | . 2 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → (∃𝑓 ∈ (𝑅 ↑m 𝑉)(𝑆‘𝑓) = 𝐹 → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
33 | 14, 32 | mpan9 506 | 1 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ (𝐶 ∖ 𝑉)) → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ⊆ wss 3963 ∅c0 4339 ifcif 4531 ran crn 5690 “ cima 5692 Fun wfun 6557 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ↑pm cpm 8866 〈“cs1 14630 mCNcmcn 35445 mVRcmvar 35446 mRExcmrex 35451 mRSubstcmrsub 35455 |
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 |
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-op 4638 df-uni 4913 df-int 4952 df-iun 4998 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-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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 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-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-gsum 17489 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-frmd 18875 df-mrex 35471 df-mrsub 35475 |
This theorem is referenced by: elmrsubrn 35505 mrsubco 35506 mrsubvrs 35507 |
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