Theorem 2wlkdlem6 26827

Description: Lemma 6 for 2wlkd 26832. (Contributed by AV, 23-Jan-2021.)

Hypotheses

Ref	Expression
2wlkd.p	⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩
2wlkd.f	⊢ 𝐹 = ⟨“𝐽𝐾”⟩
2wlkd.s	⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
2wlkd.n	⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))
2wlkd.e	⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))

Assertion

Ref	Expression
2wlkdlem6	⊢ (𝜑 → (𝐵 ∈ (𝐼‘𝐽) ∧ 𝐵 ∈ (𝐼‘𝐾)))

Proof of Theorem 2wlkdlem6

Step	Hyp	Ref	Expression
1		2wlkd.e	. 2 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))
2		prcom 4267	. . . . . . . . 9 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
3	2	sseq1i 3629	. . . . . . . 8 ⊢ ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ↔ {𝐵, 𝐴} ⊆ (𝐼‘𝐽))
4	3	biimpi 206	. . . . . . 7 ⊢ ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) → {𝐵, 𝐴} ⊆ (𝐼‘𝐽))
5	4	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → {𝐵, 𝐴} ⊆ (𝐼‘𝐽))
6		2wlkd.s	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
7	6	simp2d 1074	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
8	6	simp1d 1073	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
9	8	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → 𝐴 ∈ 𝑉)
10		prssg 4350	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ((𝐵 ∈ (𝐼‘𝐽) ∧ 𝐴 ∈ (𝐼‘𝐽)) ↔ {𝐵, 𝐴} ⊆ (𝐼‘𝐽)))
11	7, 9, 10	syl2an2r 876	. . . . . 6 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → ((𝐵 ∈ (𝐼‘𝐽) ∧ 𝐴 ∈ (𝐼‘𝐽)) ↔ {𝐵, 𝐴} ⊆ (𝐼‘𝐽)))
12	5, 11	mpbird 247	. . . . 5 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → (𝐵 ∈ (𝐼‘𝐽) ∧ 𝐴 ∈ (𝐼‘𝐽)))
13	12	simpld 475	. . . 4 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → 𝐵 ∈ (𝐼‘𝐽))
14	13	ex 450	. . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) → 𝐵 ∈ (𝐼‘𝐽)))
15		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → {𝐵, 𝐶} ⊆ (𝐼‘𝐾))
16	6	simp3d 1075	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
17	16	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → 𝐶 ∈ 𝑉)
18		prssg 4350	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐵 ∈ (𝐼‘𝐾) ∧ 𝐶 ∈ (𝐼‘𝐾)) ↔ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))
19	7, 17, 18	syl2an2r 876	. . . . . 6 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → ((𝐵 ∈ (𝐼‘𝐾) ∧ 𝐶 ∈ (𝐼‘𝐾)) ↔ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))
20	15, 19	mpbird 247	. . . . 5 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → (𝐵 ∈ (𝐼‘𝐾) ∧ 𝐶 ∈ (𝐼‘𝐾)))
21	20	simpld 475	. . . 4 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → 𝐵 ∈ (𝐼‘𝐾))
22	21	ex 450	. . 3 ⊢ (𝜑 → ({𝐵, 𝐶} ⊆ (𝐼‘𝐾) → 𝐵 ∈ (𝐼‘𝐾)))
23	14, 22	anim12d 586	. 2 ⊢ (𝜑 → (({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → (𝐵 ∈ (𝐼‘𝐽) ∧ 𝐵 ∈ (𝐼‘𝐾))))
24	1, 23	mpd 15	1 ⊢ (𝜑 → (𝐵 ∈ (𝐼‘𝐽) ∧ 𝐵 ∈ (𝐼‘𝐾)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ⊆ wss 3574  {cpr 4179  ‘cfv 5888  ⟨“cs2 13586  ⟨“cs3 13587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-pr 4180

This theorem is referenced by:  2wlkdlem7  26828