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Mirrors > Home > MPE Home > Th. List > 2trllemH | Structured version Visualization version GIF version |
Description: Lemma 3 for constr2trl 26129. (Contributed by Alexander van der Vekens, 31-Jan-2018.) |
Ref | Expression |
---|---|
2trlX.i | ⊢ (𝐼 ∈ 𝑈 ∧ 𝐽 ∈ 𝑊) |
2trlX.f | ⊢ 𝐹 = {〈0, 𝐼〉, 〈1, 𝐽〉} |
Ref | Expression |
---|---|
2trllemH | ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐵 ∈ 𝑉) ∧ ((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → 𝐹:(0..^(#‘𝐹))⟶dom 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 9913 | . . . . . 6 ⊢ 0 ∈ V | |
2 | 1ex 9914 | . . . . . 6 ⊢ 1 ∈ V | |
3 | 1, 2 | pm3.2i 470 | . . . . 5 ⊢ (0 ∈ V ∧ 1 ∈ V) |
4 | 2trlX.i | . . . . 5 ⊢ (𝐼 ∈ 𝑈 ∧ 𝐽 ∈ 𝑊) | |
5 | 0ne1 10965 | . . . . 5 ⊢ 0 ≠ 1 | |
6 | 3, 4, 5 | 3pm3.2i 1232 | . . . 4 ⊢ ((0 ∈ V ∧ 1 ∈ V) ∧ (𝐼 ∈ 𝑈 ∧ 𝐽 ∈ 𝑊) ∧ 0 ≠ 1) |
7 | fprg 6327 | . . . . 5 ⊢ (((0 ∈ V ∧ 1 ∈ V) ∧ (𝐼 ∈ 𝑈 ∧ 𝐽 ∈ 𝑊) ∧ 0 ≠ 1) → {〈0, 𝐼〉, 〈1, 𝐽〉}:{0, 1}⟶{𝐼, 𝐽}) | |
8 | 2trlX.f | . . . . . . . 8 ⊢ 𝐹 = {〈0, 𝐼〉, 〈1, 𝐽〉} | |
9 | 4, 8 | 2trllemB 26081 | . . . . . . 7 ⊢ (0..^(#‘𝐹)) = {0, 1} |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (((0 ∈ V ∧ 1 ∈ V) ∧ (𝐼 ∈ 𝑈 ∧ 𝐽 ∈ 𝑊) ∧ 0 ≠ 1) → (0..^(#‘𝐹)) = {0, 1}) |
11 | 10 | feq2d 5944 | . . . . 5 ⊢ (((0 ∈ V ∧ 1 ∈ V) ∧ (𝐼 ∈ 𝑈 ∧ 𝐽 ∈ 𝑊) ∧ 0 ≠ 1) → ({〈0, 𝐼〉, 〈1, 𝐽〉}:(0..^(#‘𝐹))⟶{𝐼, 𝐽} ↔ {〈0, 𝐼〉, 〈1, 𝐽〉}:{0, 1}⟶{𝐼, 𝐽})) |
12 | 7, 11 | mpbird 246 | . . . 4 ⊢ (((0 ∈ V ∧ 1 ∈ V) ∧ (𝐼 ∈ 𝑈 ∧ 𝐽 ∈ 𝑊) ∧ 0 ≠ 1) → {〈0, 𝐼〉, 〈1, 𝐽〉}:(0..^(#‘𝐹))⟶{𝐼, 𝐽}) |
13 | 6, 12 | mp1i 13 | . . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐵 ∈ 𝑉) ∧ ((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → {〈0, 𝐼〉, 〈1, 𝐽〉}:(0..^(#‘𝐹))⟶{𝐼, 𝐽}) |
14 | 2trllemF 26079 | . . . . . . . . 9 ⊢ (((𝐸‘𝐼) = {𝐴, 𝐵} ∧ 𝐵 ∈ 𝑉) → 𝐼 ∈ dom 𝐸) | |
15 | 14 | adantlr 747 | . . . . . . . 8 ⊢ ((((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶}) ∧ 𝐵 ∈ 𝑉) → 𝐼 ∈ dom 𝐸) |
16 | prcom 4211 | . . . . . . . . . . . 12 ⊢ {𝐵, 𝐶} = {𝐶, 𝐵} | |
17 | 16 | eqeq2i 2622 | . . . . . . . . . . 11 ⊢ ((𝐸‘𝐽) = {𝐵, 𝐶} ↔ (𝐸‘𝐽) = {𝐶, 𝐵}) |
18 | 17 | biimpi 205 | . . . . . . . . . 10 ⊢ ((𝐸‘𝐽) = {𝐵, 𝐶} → (𝐸‘𝐽) = {𝐶, 𝐵}) |
19 | 18 | adantl 481 | . . . . . . . . 9 ⊢ (((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶}) → (𝐸‘𝐽) = {𝐶, 𝐵}) |
20 | 2trllemF 26079 | . . . . . . . . 9 ⊢ (((𝐸‘𝐽) = {𝐶, 𝐵} ∧ 𝐵 ∈ 𝑉) → 𝐽 ∈ dom 𝐸) | |
21 | 19, 20 | sylan 487 | . . . . . . . 8 ⊢ ((((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶}) ∧ 𝐵 ∈ 𝑉) → 𝐽 ∈ dom 𝐸) |
22 | 15, 21 | jca 553 | . . . . . . 7 ⊢ ((((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶}) ∧ 𝐵 ∈ 𝑉) → (𝐼 ∈ dom 𝐸 ∧ 𝐽 ∈ dom 𝐸)) |
23 | 22 | expcom 450 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶}) → (𝐼 ∈ dom 𝐸 ∧ 𝐽 ∈ dom 𝐸))) |
24 | 23 | 3ad2ant3 1077 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐵 ∈ 𝑉) → (((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶}) → (𝐼 ∈ dom 𝐸 ∧ 𝐽 ∈ dom 𝐸))) |
25 | 24 | imp 444 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐵 ∈ 𝑉) ∧ ((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → (𝐼 ∈ dom 𝐸 ∧ 𝐽 ∈ dom 𝐸)) |
26 | prssg 4290 | . . . . 5 ⊢ ((𝐼 ∈ 𝑈 ∧ 𝐽 ∈ 𝑊) → ((𝐼 ∈ dom 𝐸 ∧ 𝐽 ∈ dom 𝐸) ↔ {𝐼, 𝐽} ⊆ dom 𝐸)) | |
27 | 4, 26 | ax-mp 5 | . . . 4 ⊢ ((𝐼 ∈ dom 𝐸 ∧ 𝐽 ∈ dom 𝐸) ↔ {𝐼, 𝐽} ⊆ dom 𝐸) |
28 | 25, 27 | sylib 207 | . . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐵 ∈ 𝑉) ∧ ((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → {𝐼, 𝐽} ⊆ dom 𝐸) |
29 | 13, 28 | fssd 5970 | . 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐵 ∈ 𝑉) ∧ ((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → {〈0, 𝐼〉, 〈1, 𝐽〉}:(0..^(#‘𝐹))⟶dom 𝐸) |
30 | 8 | feq1i 5949 | . 2 ⊢ (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ↔ {〈0, 𝐼〉, 〈1, 𝐽〉}:(0..^(#‘𝐹))⟶dom 𝐸) |
31 | 29, 30 | sylibr 223 | 1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐵 ∈ 𝑉) ∧ ((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → 𝐹:(0..^(#‘𝐹))⟶dom 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 {cpr 4127 〈cop 4131 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 ..^cfzo 12334 #chash 12979 |
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 ax-pre-mulgt0 9892 |
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-rmo 2904 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-int 4411 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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 |
This theorem is referenced by: constr2wlk 26128 |
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