Theorem 2wlkdlem6 30024

Description: Lemma 6 for 2wlkd 30029. (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 4671	. . . . . . . 8 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
3	2	sseq1i 3950	. . . . . . 7 ⊢ ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ↔ {𝐵, 𝐴} ⊆ (𝐼‘𝐽))
4	3	bilani 505	. . . . . 6 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → {𝐵, 𝐴} ⊆ (𝐼‘𝐽))
5		2wlkd.s	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
6	5	simp2d 1149	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7	5	simp1d 1148	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8	7	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → 𝐴 ∈ 𝑉)
9		prssg 4757	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ((𝐵 ∈ (𝐼‘𝐽) ∧ 𝐴 ∈ (𝐼‘𝐽)) ↔ {𝐵, 𝐴} ⊆ (𝐼‘𝐽)))
10	6, 8, 9	syl2an2r 691	. . . . . 6 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → ((𝐵 ∈ (𝐼‘𝐽) ∧ 𝐴 ∈ (𝐼‘𝐽)) ↔ {𝐵, 𝐴} ⊆ (𝐼‘𝐽)))
11	4, 10	mpbird 258	. . . . 5 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → (𝐵 ∈ (𝐼‘𝐽) ∧ 𝐴 ∈ (𝐼‘𝐽)))
12	11	simpld 495	. . . 4 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → 𝐵 ∈ (𝐼‘𝐽))
13	12	ex 413	. . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) → 𝐵 ∈ (𝐼‘𝐽)))
14		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → {𝐵, 𝐶} ⊆ (𝐼‘𝐾))
15	5	simp3d 1150	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
16	15	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → 𝐶 ∈ 𝑉)
17		prssg 4757	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐵 ∈ (𝐼‘𝐾) ∧ 𝐶 ∈ (𝐼‘𝐾)) ↔ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))
18	6, 16, 17	syl2an2r 691	. . . . . 6 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → ((𝐵 ∈ (𝐼‘𝐾) ∧ 𝐶 ∈ (𝐼‘𝐾)) ↔ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))
19	14, 18	mpbird 258	. . . . 5 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → (𝐵 ∈ (𝐼‘𝐾) ∧ 𝐶 ∈ (𝐼‘𝐾)))
20	19	simpld 495	. . . 4 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → 𝐵 ∈ (𝐼‘𝐾))
21	20	ex 413	. . 3 ⊢ (𝜑 → ({𝐵, 𝐶} ⊆ (𝐼‘𝐾) → 𝐵 ∈ (𝐼‘𝐾)))
22	13, 21	anim12d 615	. 2 ⊢ (𝜑 → (({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → (𝐵 ∈ (𝐼‘𝐽) ∧ 𝐵 ∈ (𝐼‘𝐾))))
23	1, 22	mpd 15	1 ⊢ (𝜑 → (𝐵 ∈ (𝐼‘𝐽) ∧ 𝐵 ∈ (𝐼‘𝐾)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935   ⊆ wss 3890  {cpr 4564  ‘cfv 6492  ⟨“cs2 14801  ⟨“cs3 14802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-un 3895  df-ss 3907  df-sn 4563  df-pr 4565

This theorem is referenced by:  2wlkdlem7  30025