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Mirrors > Home > MPE Home > Th. List > 1pthdlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for 1pthd 29971. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) |
Ref | Expression |
---|---|
1wlkd.p | ⊢ 𝑃 = 〈“𝑋𝑌”〉 |
1wlkd.f | ⊢ 𝐹 = 〈“𝐽”〉 |
Ref | Expression |
---|---|
1pthdlem1 | ⊢ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fun0 6621 | . 2 ⊢ Fun ∅ | |
2 | 1wlkd.f | . . . . . . . . . . 11 ⊢ 𝐹 = 〈“𝐽”〉 | |
3 | 2 | fveq2i 6903 | . . . . . . . . . 10 ⊢ (♯‘𝐹) = (♯‘〈“𝐽”〉) |
4 | s1len 14594 | . . . . . . . . . 10 ⊢ (♯‘〈“𝐽”〉) = 1 | |
5 | 3, 4 | eqtri 2755 | . . . . . . . . 9 ⊢ (♯‘𝐹) = 1 |
6 | 5 | oveq2i 7435 | . . . . . . . 8 ⊢ (1..^(♯‘𝐹)) = (1..^1) |
7 | fzo0 13694 | . . . . . . . 8 ⊢ (1..^1) = ∅ | |
8 | 6, 7 | eqtri 2755 | . . . . . . 7 ⊢ (1..^(♯‘𝐹)) = ∅ |
9 | 8 | reseq2i 5984 | . . . . . 6 ⊢ (𝑃 ↾ (1..^(♯‘𝐹))) = (𝑃 ↾ ∅) |
10 | res0 5991 | . . . . . 6 ⊢ (𝑃 ↾ ∅) = ∅ | |
11 | 9, 10 | eqtri 2755 | . . . . 5 ⊢ (𝑃 ↾ (1..^(♯‘𝐹))) = ∅ |
12 | 11 | cnveqi 5879 | . . . 4 ⊢ ◡(𝑃 ↾ (1..^(♯‘𝐹))) = ◡∅ |
13 | cnv0 6148 | . . . 4 ⊢ ◡∅ = ∅ | |
14 | 12, 13 | eqtri 2755 | . . 3 ⊢ ◡(𝑃 ↾ (1..^(♯‘𝐹))) = ∅ |
15 | 14 | funeqi 6577 | . 2 ⊢ (Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ↔ Fun ∅) |
16 | 1, 15 | mpbir 230 | 1 ⊢ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∅c0 4324 ◡ccnv 5679 ↾ cres 5682 Fun wfun 6545 ‘cfv 6551 (class class class)co 7424 1c1 11145 ..^cfzo 13665 ♯chash 14327 〈“cs1 14583 〈“cs2 14830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-n0 12509 df-z 12595 df-uz 12859 df-fz 13523 df-fzo 13666 df-hash 14328 df-s1 14584 |
This theorem is referenced by: 1pthd 29971 |
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