Theorem mreexexlemd 17823

Description: This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 17827. (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 781	. . . . . . . . . . 11 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → 𝑢 = 𝑓)
6	5	breq1d 5112	. . . . . . . . . 10 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (𝑢 ≈ 𝐾 ↔ 𝑓 ≈ 𝐾))
7		simpr 490	. . . . . . . . . . 11 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → 𝑣 = 𝑔)
8	7	breq1d 5112	. . . . . . . . . 10 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (𝑣 ≈ 𝐾 ↔ 𝑔 ≈ 𝐾))
9	6, 8	orbi12d 932	. . . . . . . . 9 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → ((𝑢 ≈ 𝐾 ∨ 𝑣 ≈ 𝐾) ↔ (𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾)))
10		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → 𝑡 = ℎ)
11	7, 10	uneq12d 4168	. . . . . . . . . . 11 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (𝑣 ∪ 𝑡) = (𝑔 ∪ ℎ))
12	11	fveq2d 6889	. . . . . . . . . 10 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (𝑁‘(𝑣 ∪ 𝑡)) = (𝑁‘(𝑔 ∪ ℎ)))
13	5, 12	sseq12d 3941	. . . . . . . . 9 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (𝑢 ⊆ (𝑁‘(𝑣 ∪ 𝑡)) ↔ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ))))
14	5, 10	uneq12d 4168	. . . . . . . . . 10 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → (𝑢 ∪ 𝑡) = (𝑓 ∪ ℎ))
15	14	eleq1d 2845	. . . . . . . . 9 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → ((𝑢 ∪ 𝑡) ∈ 𝐼 ↔ (𝑓 ∪ ℎ) ∈ 𝐼))
16	9, 13, 15	3anbi123d 1464	. . . . . . . 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 788	. . . . . . . . . . 11 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → 𝑢 = 𝑓)
18		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → 𝑖 = 𝑗)
19	17, 18	breq12d 5115	. . . . . . . . . 10 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → (𝑢 ≈ 𝑖 ↔ 𝑓 ≈ 𝑗))
20		simplll 787	. . . . . . . . . . . 12 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → 𝑡 = ℎ)
21	18, 20	uneq12d 4168	. . . . . . . . . . 11 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → (𝑖 ∪ 𝑡) = (𝑗 ∪ ℎ))
22	21	eleq1d 2845	. . . . . . . . . 10 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → ((𝑖 ∪ 𝑡) ∈ 𝐼 ↔ (𝑗 ∪ ℎ) ∈ 𝐼))
23	19, 22	anbi12d 644	. . . . . . . . 9 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → ((𝑢 ≈ 𝑖 ∧ (𝑖 ∪ 𝑡) ∈ 𝐼) ↔ (𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼)))
24		simplr 781	. . . . . . . . . 10 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → 𝑣 = 𝑔)
25	24	pweqd 4573	. . . . . . . . 9 ⊢ ((((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) ∧ 𝑖 = 𝑗) → 𝒫 𝑣 = 𝒫 𝑔)
26	23, 25	cbvrexdva2 3337	. . . . . . . 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 347	. . . . . . 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 488	. . . . . . . . . 10 ⊢ ((𝑡 = ℎ ∧ 𝑢 = 𝑓) → 𝑡 = ℎ)
29	28	difeq2d 4063	. . . . . . . . 9 ⊢ ((𝑡 = ℎ ∧ 𝑢 = 𝑓) → (𝑋 ∖ 𝑡) = (𝑋 ∖ ℎ))
30	29	pweqd 4573	. . . . . . . 8 ⊢ ((𝑡 = ℎ ∧ 𝑢 = 𝑓) → 𝒫 (𝑋 ∖ 𝑡) = 𝒫 (𝑋 ∖ ℎ))
31	30	adantr 486	. . . . . . 7 ⊢ (((𝑡 = ℎ ∧ 𝑢 = 𝑓) ∧ 𝑣 = 𝑔) → 𝒫 (𝑋 ∖ 𝑡) = 𝒫 (𝑋 ∖ ℎ))
32	27, 31	cbvraldva2 3336	. . . . . 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 3336	. . . . 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 2068	. . . 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 221	. . 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 4177	. . . . . 6 ⊢ 𝐻 ⊆ (𝐹 ∪ 𝐻)
37	36	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐻 ⊆ (𝐹 ∪ 𝐻))
38	3, 37	ssexd 5285	. . . 4 ⊢ (𝜑 → 𝐻 ∈ V)
39		mreexexlemd.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
40	39	difexd 5292	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∖ 𝐻) ∈ V)
41		mreexexlemd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))
42	40, 41	sselpwd 5289	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝒫 (𝑋 ∖ 𝐻))
43	42	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ℎ = 𝐻) → 𝐹 ∈ 𝒫 (𝑋 ∖ 𝐻))
44		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ = 𝐻) → ℎ = 𝐻)
45	44	difeq2d 4063	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ = 𝐻) → (𝑋 ∖ ℎ) = (𝑋 ∖ 𝐻))
46	45	pweqd 4573	. . . . . 6 ⊢ ((𝜑 ∧ ℎ = 𝐻) → 𝒫 (𝑋 ∖ ℎ) = 𝒫 (𝑋 ∖ 𝐻))
47	43, 46	eleqtrrd 2863	. . . . 5 ⊢ ((𝜑 ∧ ℎ = 𝐻) → 𝐹 ∈ 𝒫 (𝑋 ∖ ℎ))
48		mreexexlemd.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻))
49	40, 48	sselpwd 5289	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝒫 (𝑋 ∖ 𝐻))
50	49	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) → 𝐺 ∈ 𝒫 (𝑋 ∖ 𝐻))
51	46	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) → 𝒫 (𝑋 ∖ ℎ) = 𝒫 (𝑋 ∖ 𝐻))
52	50, 51	eleqtrrd 2863	. . . . . 6 ⊢ (((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) → 𝐺 ∈ 𝒫 (𝑋 ∖ ℎ))
53		simplr 781	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
54	53	breq1d 5112	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓 ≈ 𝐾 ↔ 𝐹 ≈ 𝐾))
55		simpr 490	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
56	55	breq1d 5112	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔 ≈ 𝐾 ↔ 𝐺 ≈ 𝐾))
57	54, 56	orbi12d 932	. . . . . . . 8 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑓 ≈ 𝐾 ∨ 𝑔 ≈ 𝐾) ↔ (𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾)))
58		simpllr 788	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ℎ = 𝐻)
59	55, 58	uneq12d 4168	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔 ∪ ℎ) = (𝐺 ∪ 𝐻))
60	59	fveq2d 6889	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑁‘(𝑔 ∪ ℎ)) = (𝑁‘(𝐺 ∪ 𝐻)))
61	53, 60	sseq12d 3941	. . . . . . . 8 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ↔ 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))))
62	53, 58	uneq12d 4168	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓 ∪ ℎ) = (𝐹 ∪ 𝐻))
63	62	eleq1d 2845	. . . . . . . 8 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑓 ∪ ℎ) ∈ 𝐼 ↔ (𝐹 ∪ 𝐻) ∈ 𝐼))
64	57, 61, 63	3anbi123d 1464	. . . . . . 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)))
65	55	pweqd 4573	. . . . . . . 8 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝒫 𝑔 = 𝒫 𝐺)
66	53	breq1d 5112	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓 ≈ 𝑗 ↔ 𝐹 ≈ 𝑗))
67	58	uneq2d 4167	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑗 ∪ ℎ) = (𝑗 ∪ 𝐻))
68	67	eleq1d 2845	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑗 ∪ ℎ) ∈ 𝐼 ↔ (𝑗 ∪ 𝐻) ∈ 𝐼))
69	66, 68	anbi12d 644	. . . . . . . 8 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼) ↔ (𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)))
70	65, 69	rexeqbidv 3335	. . . . . . 7 ⊢ ((((𝜑 ∧ ℎ = 𝐻) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼) ↔ ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)))
71	64, 70	imbi12d 347	. . . . . 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))))
72	52, 71	rspcdv 3568	. . . . 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))))
73	47, 72	rspcimdv 3566	. . . 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))))
74	38, 73	spcimdv 3547	. . 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))))
75	35, 74	mpd 16	. 2 ⊢ (𝜑 → (((𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾) ∧ 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)) ∧ (𝐹 ∪ 𝐻) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)))
76	1, 2, 3, 75	mp3and 1493	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   ∖ cdif 4045   ∪ cun 4149  𝒫 cpw 4556   class class class wbr 5102  ‘cfv 6538   ≈ cen 8973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546

This theorem is used by:  mreexexlem4d  17826  mreexexd  17827