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Mirrors > Home > MPE Home > Th. List > opsrbaslemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of opsrbaslem 19298 as of 9-Sep-2021. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
opsrbas.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
opsrbas.o | ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) |
opsrbas.t | ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) |
opsrbaslemOLD.1 | ⊢ 𝐸 = Slot 𝑁 |
opsrbaslemOLD.2 | ⊢ 𝑁 ∈ ℕ |
opsrbaslemOLD.3 | ⊢ 𝑁 < 10 |
Ref | Expression |
---|---|
opsrbaslemOLD | ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opsrbas.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
2 | opsrbas.o | . . . . 5 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) | |
3 | eqid 2610 | . . . . 5 ⊢ (le‘𝑂) = (le‘𝑂) | |
4 | simprl 790 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝐼 ∈ V) | |
5 | simprr 792 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑅 ∈ V) | |
6 | opsrbas.t | . . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) | |
7 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑇 ⊆ (𝐼 × 𝐼)) |
8 | 1, 2, 3, 4, 5, 7 | opsrval2 19297 | . . . 4 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑂 = (𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉)) |
9 | 8 | fveq2d 6107 | . . 3 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑂) = (𝐸‘(𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉))) |
10 | opsrbaslemOLD.1 | . . . . 5 ⊢ 𝐸 = Slot 𝑁 | |
11 | opsrbaslemOLD.2 | . . . . 5 ⊢ 𝑁 ∈ ℕ | |
12 | 10, 11 | ndxid 15716 | . . . 4 ⊢ 𝐸 = Slot (𝐸‘ndx) |
13 | 11 | nnrei 10906 | . . . . . 6 ⊢ 𝑁 ∈ ℝ |
14 | opsrbaslemOLD.3 | . . . . . 6 ⊢ 𝑁 < 10 | |
15 | 13, 14 | ltneii 10029 | . . . . 5 ⊢ 𝑁 ≠ 10 |
16 | 10, 11 | ndxarg 15715 | . . . . . 6 ⊢ (𝐸‘ndx) = 𝑁 |
17 | plendxOLD 15871 | . . . . . 6 ⊢ (le‘ndx) = 10 | |
18 | 16, 17 | neeq12i 2848 | . . . . 5 ⊢ ((𝐸‘ndx) ≠ (le‘ndx) ↔ 𝑁 ≠ 10) |
19 | 15, 18 | mpbir 220 | . . . 4 ⊢ (𝐸‘ndx) ≠ (le‘ndx) |
20 | 12, 19 | setsnid 15743 | . . 3 ⊢ (𝐸‘𝑆) = (𝐸‘(𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉)) |
21 | 9, 20 | syl6reqr 2663 | . 2 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑆) = (𝐸‘𝑂)) |
22 | 0fv 6137 | . . . . . . 7 ⊢ (∅‘𝑇) = ∅ | |
23 | 22 | eqcomi 2619 | . . . . . 6 ⊢ ∅ = (∅‘𝑇) |
24 | reldmpsr 19182 | . . . . . . 7 ⊢ Rel dom mPwSer | |
25 | 24 | ovprc 6581 | . . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅) |
26 | reldmopsr 19294 | . . . . . . . 8 ⊢ Rel dom ordPwSer | |
27 | 26 | ovprc 6581 | . . . . . . 7 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 ordPwSer 𝑅) = ∅) |
28 | 27 | fveq1d 6105 | . . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 ordPwSer 𝑅)‘𝑇) = (∅‘𝑇)) |
29 | 23, 25, 28 | 3eqtr4a 2670 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ((𝐼 ordPwSer 𝑅)‘𝑇)) |
30 | 29 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐼 mPwSer 𝑅) = ((𝐼 ordPwSer 𝑅)‘𝑇)) |
31 | 30, 1, 2 | 3eqtr4g 2669 | . . 3 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑆 = 𝑂) |
32 | 31 | fveq2d 6107 | . 2 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑆) = (𝐸‘𝑂)) |
33 | 21, 32 | pm2.61dan 828 | 1 ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 〈cop 4131 class class class wbr 4583 × cxp 5036 ‘cfv 5804 (class class class)co 6549 < clt 9953 ℕcn 10897 10c10 10955 ndxcnx 15692 sSet csts 15693 Slot cslot 15694 lecple 15775 mPwSer cmps 19172 ordPwSer copws 19176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-10OLD 10964 df-dec 11370 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ple 15788 df-psr 19177 df-opsr 19181 |
This theorem is referenced by: (None) |
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