Theorem fiunlem 7966

Description: Lemma for fiun 7967 and f1iun 7968. Formerly part of f1iun 7968. (Contributed by AV, 6-Oct-2023.)

Hypothesis

Ref	Expression
fiun.1	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

Ref	Expression
fiunlem	⊢ (((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))

Distinct variable groups:   𝑣,𝐴,𝑥,𝑧   𝑦,𝐴,𝑣   𝑣,𝐵,𝑦   𝑧,𝐵   𝑣,𝐶,𝑥   𝑣,𝐷   𝑣,𝑆   𝑣,𝑢,𝑦   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑢)   𝐵(𝑥,𝑢)   𝐶(𝑦,𝑧,𝑢)   𝐷(𝑥,𝑦,𝑧,𝑢)   𝑆(𝑥,𝑦,𝑧,𝑢)

Proof of Theorem fiunlem

Step	Hyp	Ref	Expression
1		vex 3484	. . . 4 ⊢ 𝑣 ∈ V
2		eqeq1 2741	. . . . 5 ⊢ (𝑧 = 𝑣 → (𝑧 = 𝐵 ↔ 𝑣 = 𝐵))
3	2	rexbidv 3179	. . . 4 ⊢ (𝑧 = 𝑣 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑣 = 𝐵))
4	1, 3	elab 3679	. . 3 ⊢ (𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑣 = 𝐵)
5		fiun.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
6	5	eqeq2d 2748	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑣 = 𝐵 ↔ 𝑣 = 𝐶))
7	6	cbvrexvw 3238	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑣 = 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑣 = 𝐶)
8		r19.29 3114	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵) ∧ ∃𝑦 ∈ 𝐴 𝑣 = 𝐶) → ∃𝑦 ∈ 𝐴 ((𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵) ∧ 𝑣 = 𝐶))
9		sseq12 4011	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝐵 ∧ 𝑣 = 𝐶) → (𝑢 ⊆ 𝑣 ↔ 𝐵 ⊆ 𝐶))
10	9	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑣 = 𝐶 ∧ 𝑢 = 𝐵) → (𝑢 ⊆ 𝑣 ↔ 𝐵 ⊆ 𝐶))
11		sseq12 4011	. . . . . . . . . . . 12 ⊢ ((𝑣 = 𝐶 ∧ 𝑢 = 𝐵) → (𝑣 ⊆ 𝑢 ↔ 𝐶 ⊆ 𝐵))
12	10, 11	orbi12d 919	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝐶 ∧ 𝑢 = 𝐵) → ((𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢) ↔ (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)))
13	12	biimprcd 250	. . . . . . . . . 10 ⊢ ((𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵) → ((𝑣 = 𝐶 ∧ 𝑢 = 𝐵) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)))
14	13	expdimp 452	. . . . . . . . 9 ⊢ (((𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵) ∧ 𝑣 = 𝐶) → (𝑢 = 𝐵 → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)))
15	14	rexlimivw 3151	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐴 ((𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵) ∧ 𝑣 = 𝐶) → (𝑢 = 𝐵 → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)))
16	15	imp 406	. . . . . . 7 ⊢ ((∃𝑦 ∈ 𝐴 ((𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵) ∧ 𝑣 = 𝐶) ∧ 𝑢 = 𝐵) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
17	8, 16	sylan 580	. . . . . 6 ⊢ (((∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵) ∧ ∃𝑦 ∈ 𝐴 𝑣 = 𝐶) ∧ 𝑢 = 𝐵) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
18	17	an32s 652	. . . . 5 ⊢ (((∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵) ∧ 𝑢 = 𝐵) ∧ ∃𝑦 ∈ 𝐴 𝑣 = 𝐶) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
19	18	adantlll 718	. . . 4 ⊢ ((((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 = 𝐵) ∧ ∃𝑦 ∈ 𝐴 𝑣 = 𝐶) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
20	7, 19	sylan2b 594	. . 3 ⊢ ((((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 = 𝐵) ∧ ∃𝑥 ∈ 𝐴 𝑣 = 𝐵) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
21	4, 20	sylan2b 594	. 2 ⊢ ((((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 = 𝐵) ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
22	21	ralrimiva 3146	1 ⊢ (((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ⟶wf 6557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968

This theorem is referenced by:  fiun  7967  f1iun  7968