Theorem 2wlkdlem6 29185

Description: Lemma 6 for 2wlkd 29190. (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 4737	. . . . . . . . 9 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
3	2	sseq1i 4011	. . . . . . . 8 ⊢ ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ↔ {𝐵, 𝐴} ⊆ (𝐼‘𝐽))
4	3	biimpi 215	. . . . . . 7 ⊢ ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) → {𝐵, 𝐴} ⊆ (𝐼‘𝐽))
5	4	adantl 483	. . . . . 6 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → {𝐵, 𝐴} ⊆ (𝐼‘𝐽))
6		2wlkd.s	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
7	6	simp2d 1144	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
8	6	simp1d 1143	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
9	8	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → 𝐴 ∈ 𝑉)
10		prssg 4823	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ((𝐵 ∈ (𝐼‘𝐽) ∧ 𝐴 ∈ (𝐼‘𝐽)) ↔ {𝐵, 𝐴} ⊆ (𝐼‘𝐽)))
11	7, 9, 10	syl2an2r 684	. . . . . 6 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → ((𝐵 ∈ (𝐼‘𝐽) ∧ 𝐴 ∈ (𝐼‘𝐽)) ↔ {𝐵, 𝐴} ⊆ (𝐼‘𝐽)))
12	5, 11	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → (𝐵 ∈ (𝐼‘𝐽) ∧ 𝐴 ∈ (𝐼‘𝐽)))
13	12	simpld 496	. . . 4 ⊢ ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) → 𝐵 ∈ (𝐼‘𝐽))
14	13	ex 414	. . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) → 𝐵 ∈ (𝐼‘𝐽)))
15		simpr 486	. . . . . 6 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → {𝐵, 𝐶} ⊆ (𝐼‘𝐾))
16	6	simp3d 1145	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
17	16	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → 𝐶 ∈ 𝑉)
18		prssg 4823	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐵 ∈ (𝐼‘𝐾) ∧ 𝐶 ∈ (𝐼‘𝐾)) ↔ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))
19	7, 17, 18	syl2an2r 684	. . . . . 6 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → ((𝐵 ∈ (𝐼‘𝐾) ∧ 𝐶 ∈ (𝐼‘𝐾)) ↔ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))
20	15, 19	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → (𝐵 ∈ (𝐼‘𝐾) ∧ 𝐶 ∈ (𝐼‘𝐾)))
21	20	simpld 496	. . . 4 ⊢ ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) → 𝐵 ∈ (𝐼‘𝐾))
22	21	ex 414	. . 3 ⊢ (𝜑 → ({𝐵, 𝐶} ⊆ (𝐼‘𝐾) → 𝐵 ∈ (𝐼‘𝐾)))
23	14, 22	anim12d 610	. 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 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941   ⊆ wss 3949  {cpr 4631  ‘cfv 6544  ⟨“cs2 14792  ⟨“cs3 14793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-sn 4630  df-pr 4632

This theorem is referenced by:  2wlkdlem7  29186