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Theorem 3wlkdlem9 27941

Description: Lemma 9 for 3wlkd 27943. (Contributed by AV, 7-Feb-2021.)

**Hypotheses**
Ref	Expression
3wlkd.p	⊢ 𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩
3wlkd.f	⊢ 𝐹 = ⟨“𝐽𝐾𝐿”⟩
3wlkd.s	⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)))
3wlkd.n	⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷))
3wlkd.e	⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿)))

**Assertion**
Ref	Expression
3wlkdlem9	⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ∧ {𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1)) ∧ {𝐶, 𝐷} ⊆ (𝐼‘(𝐹‘2))))

**Proof of Theorem 3wlkdlem9**
Step	Hyp	Ref	Expression
1		3wlkd.e	. 2 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿)))
2		3wlkd.p	. . . 4 ⊢ 𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩
3		3wlkd.f	. . . 4 ⊢ 𝐹 = ⟨“𝐽𝐾𝐿”⟩
4		3wlkd.s	. . . 4 ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)))
5		3wlkd.n	. . . 4 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷))
6	2, 3, 4, 5, 1	3wlkdlem8 27940	. . 3 ⊢ (𝜑 → ((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿))
7		fveq2 6664	. . . . . 6 ⊢ ((𝐹‘0) = 𝐽 → (𝐼‘(𝐹‘0)) = (𝐼‘𝐽))
8	7	sseq2d 3998	. . . . 5 ⊢ ((𝐹‘0) = 𝐽 → ({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ↔ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)))
9	8	3ad2ant1 1129	. . . 4 ⊢ (((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿) → ({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ↔ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)))
10		fveq2 6664	. . . . . 6 ⊢ ((𝐹‘1) = 𝐾 → (𝐼‘(𝐹‘1)) = (𝐼‘𝐾))
11	10	sseq2d 3998	. . . . 5 ⊢ ((𝐹‘1) = 𝐾 → ({𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1)) ↔ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))
12	11	3ad2ant2 1130	. . . 4 ⊢ (((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿) → ({𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1)) ↔ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))
13		fveq2 6664	. . . . . 6 ⊢ ((𝐹‘2) = 𝐿 → (𝐼‘(𝐹‘2)) = (𝐼‘𝐿))
14	13	sseq2d 3998	. . . . 5 ⊢ ((𝐹‘2) = 𝐿 → ({𝐶, 𝐷} ⊆ (𝐼‘(𝐹‘2)) ↔ {𝐶, 𝐷} ⊆ (𝐼‘𝐿)))
15	14	3ad2ant3 1131	. . . 4 ⊢ (((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿) → ({𝐶, 𝐷} ⊆ (𝐼‘(𝐹‘2)) ↔ {𝐶, 𝐷} ⊆ (𝐼‘𝐿)))
16	9, 12, 15	3anbi123d 1432	. . 3 ⊢ (((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿) → (({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ∧ {𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1)) ∧ {𝐶, 𝐷} ⊆ (𝐼‘(𝐹‘2))) ↔ ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))))
17	6, 16	syl 17	. 2 ⊢ (𝜑 → (({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ∧ {𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1)) ∧ {𝐶, 𝐷} ⊆ (𝐼‘(𝐹‘2))) ↔ ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))))
18	1, 17	mpbird 259	1 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ∧ {𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1)) ∧ {𝐶, 𝐷} ⊆ (𝐼‘(𝐹‘2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3935 {cpr 4562 ‘cfv 6349 0cc0 10531 1c1 10532 2c2 11686 ⟨“cs3 14198 ⟨“cs4 14199

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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-s2 14204 df-s3 14205 df-s4 14206

This theorem is referenced by: 3wlkdlem10 27942

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