Theorem mreexexlemd 16732

Description: This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 16736. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
mreexexlemd.1	⊢ (𝜑 → 𝑋 ∈ 𝐽)
mreexexlemd.2	⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))
mreexexlemd.3	⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻))
mreexexlemd.4	⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
mreexexlemd.5	⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)
mreexexlemd.6	⊢ (𝜑 → (𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾))
mreexexlemd.7	⊢ (𝜑 → ∀𝑡∀𝑢 ∈ 𝒫 (𝑋 ∖ 𝑡)∀𝑣 ∈ 𝒫 (𝑋 ∖ 𝑡)(((𝑢 ≈ 𝐾 ∨ 𝑣 ≈ 𝐾) ∧ 𝑢 ⊆ (𝑁‘(𝑣 ∪ 𝑡)) ∧ (𝑢 ∪ 𝑡) ∈ 𝐼) → ∃𝑖 ∈ 𝒫 𝑣(𝑢 ≈ 𝑖 ∧ (𝑖 ∪ 𝑡) ∈ 𝐼)))

Assertion

Ref	Expression
mreexexlemd	⊢ (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))

Distinct variable groups:   𝑗,𝐹   𝑗,𝐺   𝑗,𝐻   𝜑,𝑗   𝑢,𝑡,𝑣,𝑖,𝐼,𝑗   𝑡,𝐾,𝑢,𝑣   𝑡,𝑁,𝑢,𝑣   𝑡,𝑋,𝑢,𝑣

Allowed substitution hints:   𝜑(𝑣,𝑢,𝑡,𝑖)   𝐹(𝑣,𝑢,𝑡,𝑖)   𝐺(𝑣,𝑢,𝑡,𝑖)   𝐻(𝑣,𝑢,𝑡,𝑖)   𝐽(𝑣,𝑢,𝑡,𝑖,𝑗)   𝐾(𝑖,𝑗)   𝑁(𝑖,𝑗)   𝑋(𝑖,𝑗)

Proof of Theorem mreexexlemd

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mreexexlemd.6	. 2 ⊢ (𝜑 → (𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾))
2		mreexexlemd.4	. 2 ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
3		mreexexlemd.5	. 2 ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)
4		mreexexlemd.7	. . . 4 ⊢ (𝜑 → ∀𝑡∀𝑢 ∈ 𝒫 (𝑋 ∖ 𝑡)∀𝑣 ∈ 𝒫 (𝑋 ∖ 𝑡)(((𝑢 ≈ 𝐾 ∨ 𝑣 ≈ 𝐾) ∧ 𝑢 ⊆ (𝑁‘(𝑣 ∪ 𝑡)) ∧ (𝑢 ∪ 𝑡) ∈ 𝐼) → ∃𝑖 ∈ 𝒫 𝑣(𝑢 ≈ 𝑖 ∧ (𝑖 ∪ 𝑡) ∈ 𝐼)))
5		simplr 765	. . . . . . . . . . 11 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → 𝑢 = 𝑓)
6	5	breq1d 4966	. . . . . . . . . 10 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (𝑢 ≈ 𝐾 ↔ 𝑓 ≈ 𝐾))
7		simpr 485	. . . . . . . . . . 11 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → 𝑣 = 𝑔)
8	7	breq1d 4966	. . . . . . . . . 10 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (𝑣 ≈ 𝐾 ↔ 𝑔 ≈ 𝐾))
9	6, 8	orbi12d 911	. . . . . . . . 9 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → ((𝑢 ≈ 𝐾 ∨ 𝑣 ≈ 𝐾) ↔ (𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾)))
10		simpll 763	. . . . . . . . . . . 12 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → 𝑡 = ℎ)
11	7, 10	uneq12d 4056	. . . . . . . . . . 11 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (𝑣 ∪ 𝑡) = (𝑔 ∪ ℎ))
12	11	fveq2d 6534	. . . . . . . . . 10 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (𝑁‘(𝑣 ∪ 𝑡)) = (𝑁‘(𝑔 ∪ ℎ)))
13	5, 12	sseq12d 3916	. . . . . . . . 9 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (𝑢 ⊆ (𝑁‘(𝑣 ∪ 𝑡)) ↔ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ))))
14	5, 10	uneq12d 4056	. . . . . . . . . 10 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (𝑢 ∪ 𝑡) = (𝑓 ∪ ℎ))
15	14	eleq1d 2865	. . . . . . . . 9 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → ((𝑢 ∪ 𝑡) ∈ 𝐼 ↔ (𝑓 ∪ ℎ) ∈ 𝐼))
16	9, 13, 15	3anbi123d 1426	. . . . . . . 8 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (((𝑢 ≈ 𝐾 ∨ 𝑣 ≈ 𝐾) ∧ 𝑢 ⊆ (𝑁‘(𝑣 ∪ 𝑡)) ∧ (𝑢 ∪ 𝑡) ∈ 𝐼) ↔ ((𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼)))
17		simpllr 772	. . . . . . . . . . 11 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → 𝑢 = 𝑓)
18		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → 𝑖 = 𝑗)
19	17, 18	breq12d 4969	. . . . . . . . . 10 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → (𝑢 ≈ 𝑖 ↔ 𝑓 ≈ 𝑗))
20		simplll 771	. . . . . . . . . . . 12 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → 𝑡 = ℎ)
21	18, 20	uneq12d 4056	. . . . . . . . . . 11 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → (𝑖 ∪ 𝑡) = (𝑗 ∪ ℎ))
22	21	eleq1d 2865	. . . . . . . . . 10 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → ((𝑖 ∪ 𝑡) ∈ 𝐼 ↔ (𝑗 ∪ ℎ) ∈ 𝐼))
23	19, 22	anbi12d 630	. . . . . . . . 9 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → ((𝑢 ≈ 𝑖 ∧ (𝑖 ∪ 𝑡) ∈ 𝐼) ↔ (𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼)))
24		simplr 765	. . . . . . . . . 10 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → 𝑣 = 𝑔)
25	24	pweqd 4452	. . . . . . . . 9 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → 𝒫 𝑣 = 𝒫 𝑔)
26	23, 25	cbvrexdva2 3405	. . . . . . . 8 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (∃𝑖 ∈ 𝒫 𝑣(𝑢 ≈ 𝑖 ∧ (𝑖 ∪ 𝑡) ∈ 𝐼) ↔ ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼)))
27	16, 26	imbi12d 346	. . . . . . 7 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → ((((𝑢 ≈ 𝐾 ∨ 𝑣 ≈ 𝐾) ∧ 𝑢 ⊆ (𝑁‘(𝑣 ∪ 𝑡)) ∧ (𝑢 ∪ 𝑡) ∈ 𝐼) → ∃𝑖 ∈ 𝒫 𝑣(𝑢 ≈ 𝑖 ∧ (𝑖 ∪ 𝑡) ∈ 𝐼)) ↔ (((𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼))))
28		simpl 483	. . . . . . . . . 10 ⊢ ((𝑡 = ℎ ∧ 𝑢 = 𝑓) → 𝑡 = ℎ)
29	28	difeq2d 4015	. . . . . . . . 9 ⊢ ((𝑡 = ℎ ∧ 𝑢 = 𝑓) → (𝑋 ∖ 𝑡) = (𝑋 ∖ ℎ))
30	29	pweqd 4452	. . . . . . . 8 ⊢ ((𝑡 = ℎ ∧ 𝑢 = 𝑓) → 𝒫 (𝑋 ∖ 𝑡) = 𝒫 (𝑋 ∖ ℎ))
31	30	adantr 481	. . . . . . 7 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → 𝒫 (𝑋 ∖ 𝑡) = 𝒫 (𝑋 ∖ ℎ))
32	27, 31	cbvraldva2 3404	. . . . . 6 ⊢ ((𝑡 = ℎ ∧ 𝑢 = 𝑓) → (∀𝑣 ∈ 𝒫 (𝑋 ∖ 𝑡)(((𝑢 ≈ 𝐾 ∨ 𝑣 ≈ 𝐾) ∧ 𝑢 ⊆ (𝑁‘(𝑣 ∪ 𝑡)) ∧ (𝑢 ∪ 𝑡) ∈ 𝐼) → ∃𝑖 ∈ 𝒫 𝑣(𝑢 ≈ 𝑖 ∧ (𝑖 ∪ 𝑡) ∈ 𝐼)) ↔ ∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼))))
33	32, 30	cbvraldva2 3404	. . . . 5 ⊢ (𝑡 = ℎ → (∀𝑢 ∈ 𝒫 (𝑋 ∖ 𝑡)∀𝑣 ∈ 𝒫 (𝑋 ∖ 𝑡)(((𝑢 ≈ 𝐾 ∨ 𝑣 ≈ 𝐾) ∧ 𝑢 ⊆ (𝑁‘(𝑣 ∪ 𝑡)) ∧ (𝑢 ∪ 𝑡) ∈ 𝐼) → ∃𝑖 ∈ 𝒫 𝑣(𝑢 ≈ 𝑖 ∧ (𝑖 ∪ 𝑡) ∈ 𝐼)) ↔ ∀𝑓 ∈ 𝒫 (𝑋 ∖ ℎ)∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼))))
34	33	cbvalvw 2018	. . . 4 ⊢ (∀𝑡∀𝑢 ∈ 𝒫 (𝑋 ∖ 𝑡)∀𝑣 ∈ 𝒫 (𝑋 ∖ 𝑡)(((𝑢 ≈ 𝐾 ∨ 𝑣 ≈ 𝐾) ∧ 𝑢 ⊆ (𝑁‘(𝑣 ∪ 𝑡)) ∧ (𝑢 ∪ 𝑡) ∈ 𝐼) → ∃𝑖 ∈ 𝒫 𝑣(𝑢 ≈ 𝑖 ∧ (𝑖 ∪ 𝑡) ∈ 𝐼)) ↔ ∀ℎ∀𝑓 ∈ 𝒫 (𝑋 ∖ ℎ)∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼)))
35	4, 34	sylib 219	. . 3 ⊢ (𝜑 → ∀ℎ∀𝑓 ∈ 𝒫 (𝑋 ∖ ℎ)∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼)))
36		ssun2 4065	. . . . . 6 ⊢ 𝐻 ⊆ (𝐹 ∪ 𝐻)
37	36	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐻 ⊆ (𝐹 ∪ 𝐻))
38	3, 37	ssexd 5112	. . . 4 ⊢ (𝜑 → 𝐻 ∈ V)
39		mreexexlemd.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
40		difexg 5115	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐽 → (𝑋 ∖ 𝐻) ∈ V)
41	39, 40	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∖ 𝐻) ∈ V)
42		mreexexlemd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))
43	41, 42	sselpwd 5114	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝒫 (𝑋 ∖ 𝐻))
44	43	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ℎ = 𝐻) → 𝐹 ∈ 𝒫 (𝑋 ∖ 𝐻))
45		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ = 𝐻) → ℎ = 𝐻)
46	45	difeq2d 4015	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ = 𝐻) → (𝑋 ∖ ℎ) = (𝑋 ∖ 𝐻))
47	46	pweqd 4452	. . . . . 6 ⊢ ((𝜑 ∧ ℎ = 𝐻) → 𝒫 (𝑋 ∖ ℎ) = 𝒫 (𝑋 ∖ 𝐻))
48	44, 47	eleqtrrd 2884	. . . . 5 ⊢ ((𝜑 ∧ ℎ = 𝐻) → 𝐹 ∈ 𝒫 (𝑋 ∖ ℎ))
49		mreexexlemd.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻))
50	41, 49	sselpwd 5114	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝒫 (𝑋 ∖ 𝐻))
51	50	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) → 𝐺 ∈ 𝒫 (𝑋 ∖ 𝐻))
52	47	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) → 𝒫 (𝑋 ∖ ℎ) = 𝒫 (𝑋 ∖ 𝐻))
53	51, 52	eleqtrrd 2884	. . . . . 6 ⊢ (((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) → 𝐺 ∈ 𝒫 (𝑋 ∖ ℎ))
54		simplr 765	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
55	54	breq1d 4966	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓 ≈ 𝐾 ↔ 𝐹 ≈ 𝐾))
56		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
57	56	breq1d 4966	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔 ≈ 𝐾 ↔ 𝐺 ≈ 𝐾))
58	55, 57	orbi12d 911	. . . . . . . 8 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾) ↔ (𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾)))
59		simpllr 772	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ℎ = 𝐻)
60	56, 59	uneq12d 4056	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔 ∪ ℎ) = (𝐺 ∪ 𝐻))
61	60	fveq2d 6534	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑁‘(𝑔 ∪ ℎ)) = (𝑁‘(𝐺 ∪ 𝐻)))
62	54, 61	sseq12d 3916	. . . . . . . 8 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ↔ 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))))
63	54, 59	uneq12d 4056	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓 ∪ ℎ) = (𝐹 ∪ 𝐻))
64	63	eleq1d 2865	. . . . . . . 8 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑓 ∪ ℎ) ∈ 𝐼 ↔ (𝐹 ∪ 𝐻) ∈ 𝐼))
65	58, 62, 64	3anbi123d 1426	. . . . . . 7 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (((𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) ↔ ((𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾) ∧ 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)) ∧ (𝐹 ∪ 𝐻) ∈ 𝐼)))
66	56	pweqd 4452	. . . . . . . 8 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝒫 𝑔 = 𝒫 𝐺)
67	54	breq1d 4966	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓 ≈ 𝑗 ↔ 𝐹 ≈ 𝑗))
68	59	uneq2d 4055	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑗 ∪ ℎ) = (𝑗 ∪ 𝐻))
69	68	eleq1d 2865	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑗 ∪ ℎ) ∈ 𝐼 ↔ (𝑗 ∪ 𝐻) ∈ 𝐼))
70	67, 69	anbi12d 630	. . . . . . . 8 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼) ↔ (𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)))
71	66, 70	rexeqbidv 3359	. . . . . . 7 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼) ↔ ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)))
72	65, 71	imbi12d 346	. . . . . 6 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((((𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼)) ↔ (((𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾) ∧ 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)) ∧ (𝐹 ∪ 𝐻) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))))
73	53, 72	rspcdv 3557	. . . . 5 ⊢ (((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) → (∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼)) → (((𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾) ∧ 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)) ∧ (𝐹 ∪ 𝐻) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))))
74	48, 73	rspcimdv 3555	. . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐻) → (∀𝑓 ∈ 𝒫 (𝑋 ∖ ℎ)∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼)) → (((𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾) ∧ 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)) ∧ (𝐹 ∪ 𝐻) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))))
75	38, 74	spcimdv 3530	. . 3 ⊢ (𝜑 → (∀ℎ∀𝑓 ∈ 𝒫 (𝑋 ∖ ℎ)∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼)) → (((𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾) ∧ 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)) ∧ (𝐹 ∪ 𝐻) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))))
76	35, 75	mpd 15	. 2 ⊢ (𝜑 → (((𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾) ∧ 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)) ∧ (𝐹 ∪ 𝐻) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)))
77	1, 2, 3, 76	mp3and 1454	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 842   ∧ w3a 1078  ∀wal 1518   = wceq 1520   ∈ wcel 2079  ∀wral 3103  ∃wrex 3104  Vcvv 3432   ∖ cdif 3851   ∪ cun 3852   ⊆ wss 3854  𝒫 cpw 4447   class class class wbr 4956  ‘cfv 6217   ≈ cen 8344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767  ax-sep 5088

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-iota 6181  df-fv 6225

This theorem is referenced by:  mreexexlem4d  16735  mreexexd  16736