Theorem satfv1lem 33324

Description: Lemma for satfv1 33325. (Contributed by AV, 9-Nov-2023.)

Assertion

Ref	Expression
satfv1lem	⊢ ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀 ↑m ω) ∣ (𝑏‘𝐼)𝐸(𝑏‘𝐽)}} = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))})

Distinct variable groups:   𝐸,𝑏   𝐼,𝑎,𝑏,𝑧   𝐽,𝑎,𝑏,𝑧   𝑀,𝑏,𝑧   𝑁,𝑎,𝑏,𝑧

Allowed substitution hints:   𝐸(𝑧,𝑎)   𝑀(𝑎)

Proof of Theorem satfv1lem

Step	Hyp	Ref	Expression
1		fveq1 6773	. . . . . . 7 ⊢ (𝑏 = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) → (𝑏‘𝐼) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼))
2		fveq1 6773	. . . . . . 7 ⊢ (𝑏 = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) → (𝑏‘𝐽) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽))
3	1, 2	breq12d 5087	. . . . . 6 ⊢ (𝑏 = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) → ((𝑏‘𝐼)𝐸(𝑏‘𝐽) ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽)))
4	3	elrab 3624	. . . . 5 ⊢ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀 ↑m ω) ∣ (𝑏‘𝐼)𝐸(𝑏‘𝐽)} ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω) ∧ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽)))
5	4	a1i 11	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀 ↑m ω) ∣ (𝑏‘𝐼)𝐸(𝑏‘𝐽)} ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω) ∧ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽))))
6		elex 3450	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → 𝑁 ∈ V)
7	6	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → 𝑁 ∈ V)
8	7	ad2antrr 723	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝑁 ∈ V)
9		simpr 485	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝑧 ∈ 𝑀)
10	8, 9	fsnd 6759	. . . . . . . 8 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → {⟨𝑁, 𝑧⟩}:{𝑁}⟶𝑀)
11		elmapex 8636	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (𝑀 ∈ V ∧ ω ∈ V))
12	11	simpld 495	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → 𝑀 ∈ V)
13	12	adantr 481	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (𝑀 ↑m ω) ∧ 𝑧 ∈ 𝑀) → 𝑀 ∈ V)
14		snex 5354	. . . . . . . . . . 11 ⊢ {𝑁} ∈ V
15	14	a1i 11	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (𝑀 ↑m ω) ∧ 𝑧 ∈ 𝑀) → {𝑁} ∈ V)
16	13, 15	elmapd 8629	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝑀 ↑m ω) ∧ 𝑧 ∈ 𝑀) → ({⟨𝑁, 𝑧⟩} ∈ (𝑀 ↑m {𝑁}) ↔ {⟨𝑁, 𝑧⟩}:{𝑁}⟶𝑀))
17	16	adantll 711	. . . . . . . 8 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → ({⟨𝑁, 𝑧⟩} ∈ (𝑀 ↑m {𝑁}) ↔ {⟨𝑁, 𝑧⟩}:{𝑁}⟶𝑀))
18	10, 17	mpbird 256	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → {⟨𝑁, 𝑧⟩} ∈ (𝑀 ↑m {𝑁}))
19		elmapi 8637	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → 𝑎:ω⟶𝑀)
20		difssd 4067	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (ω ∖ {𝑁}) ⊆ ω)
21	19, 20	fssresd 6641	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (𝑎 ↾ (ω ∖ {𝑁})):(ω ∖ {𝑁})⟶𝑀)
22		omex 9401	. . . . . . . . . . . . 13 ⊢ ω ∈ V
23	22	difexi 5252	. . . . . . . . . . . 12 ⊢ (ω ∖ {𝑁}) ∈ V
24	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (ω ∖ {𝑁}) ∈ V)
25	12, 24	elmapd 8629	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → ((𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀 ↑m (ω ∖ {𝑁})) ↔ (𝑎 ↾ (ω ∖ {𝑁})):(ω ∖ {𝑁})⟶𝑀))
26	21, 25	mpbird 256	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀 ↑m (ω ∖ {𝑁})))
27	26	adantl 482	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) → (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀 ↑m (ω ∖ {𝑁})))
28	27	adantr 481	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀 ↑m (ω ∖ {𝑁})))
29		res0 5895	. . . . . . . . . 10 ⊢ ({⟨𝑁, 𝑧⟩} ↾ ∅) = ∅
30		res0 5895	. . . . . . . . . 10 ⊢ ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ∅) = ∅
31	29, 30	eqtr4i 2769	. . . . . . . . 9 ⊢ ({⟨𝑁, 𝑧⟩} ↾ ∅) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ∅)
32		disjdif 4405	. . . . . . . . . 10 ⊢ ({𝑁} ∩ (ω ∖ {𝑁})) = ∅
33	32	reseq2i 5888	. . . . . . . . 9 ⊢ ({⟨𝑁, 𝑧⟩} ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ({⟨𝑁, 𝑧⟩} ↾ ∅)
34	32	reseq2i 5888	. . . . . . . . 9 ⊢ ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ∅)
35	31, 33, 34	3eqtr4i 2776	. . . . . . . 8 ⊢ ({⟨𝑁, 𝑧⟩} ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ({𝑁} ∩ (ω ∖ {𝑁})))
36	35	a1i 11	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → ({⟨𝑁, 𝑧⟩} ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ({𝑁} ∩ (ω ∖ {𝑁}))))
37		elmapresaun 8668	. . . . . . 7 ⊢ (({⟨𝑁, 𝑧⟩} ∈ (𝑀 ↑m {𝑁}) ∧ (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀 ↑m (ω ∖ {𝑁})) ∧ ({⟨𝑁, 𝑧⟩} ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ({𝑁} ∩ (ω ∖ {𝑁})))) → ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ({𝑁} ∪ (ω ∖ {𝑁}))))
38	18, 28, 36, 37	syl3anc 1370	. . . . . 6 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ({𝑁} ∪ (ω ∖ {𝑁}))))
39		uncom 4087	. . . . . . . . . 10 ⊢ ({𝑁} ∪ (ω ∖ {𝑁})) = ((ω ∖ {𝑁}) ∪ {𝑁})
40		difsnid 4743	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → ((ω ∖ {𝑁}) ∪ {𝑁}) = ω)
41	39, 40	eqtr2id 2791	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → ω = ({𝑁} ∪ (ω ∖ {𝑁})))
42	41	3ad2ant1 1132	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ω = ({𝑁} ∪ (ω ∖ {𝑁})))
43	42	ad2antrr 723	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → ω = ({𝑁} ∪ (ω ∖ {𝑁})))
44	43	oveq2d 7291	. . . . . 6 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → (𝑀 ↑m ω) = (𝑀 ↑m ({𝑁} ∪ (ω ∖ {𝑁}))))
45	38, 44	eleqtrrd 2842	. . . . 5 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))
46		ibar 529	. . . . . . . 8 ⊢ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω) ∧ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽))))
47	46	adantl 482	. . . . . . 7 ⊢ (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω) ∧ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽))))
48	47	bicomd 222	. . . . . 6 ⊢ (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω) ∧ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽)) ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽)))
49		simpll1 1211	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝑁 ∈ ω)
50		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))
51	49, 9, 50	fvsnun1 7054	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) = 𝑧)
52	51	adantr 481	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) = 𝑧)
53	52, 52	breq12d 5087	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ 𝑧𝐸𝑧))
54	53	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ 𝑧𝐸𝑧))
55		fveq2 6774	. . . . . . . . . . . . . 14 ⊢ (𝐽 = 𝑁 → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁))
56	55	breq2d 5086	. . . . . . . . . . . . 13 ⊢ (𝐽 = 𝑁 → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)))
57		ifptru 1073	. . . . . . . . . . . . 13 ⊢ (𝐽 = 𝑁 → (if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)) ↔ 𝑧𝐸𝑧))
58	56, 57	bibi12d 346	. . . . . . . . . . . 12 ⊢ (𝐽 = 𝑁 → (((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽))) ↔ ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ 𝑧𝐸𝑧)))
59	58	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → (((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽))) ↔ ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ 𝑧𝐸𝑧)))
60	54, 59	mpbird 256	. . . . . . . . . 10 ⊢ ((𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽))))
61	52	adantl 482	. . . . . . . . . . . 12 ⊢ ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) = 𝑧)
62	49	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)) → 𝑁 ∈ ω)
63	62	adantl 482	. . . . . . . . . . . . 13 ⊢ ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → 𝑁 ∈ ω)
64	9	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)) → 𝑧 ∈ 𝑀)
65	64	adantl 482	. . . . . . . . . . . . 13 ⊢ ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → 𝑧 ∈ 𝑀)
66		neqne 2951	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐽 = 𝑁 → 𝐽 ≠ 𝑁)
67		simpll3 1213	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝐽 ∈ ω)
68	67	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)) → 𝐽 ∈ ω)
69	66, 68	anim12ci 614	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → (𝐽 ∈ ω ∧ 𝐽 ≠ 𝑁))
70		eldifsn 4720	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (ω ∖ {𝑁}) ↔ (𝐽 ∈ ω ∧ 𝐽 ≠ 𝑁))
71	69, 70	sylibr 233	. . . . . . . . . . . . 13 ⊢ ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → 𝐽 ∈ (ω ∖ {𝑁}))
72	63, 65, 50, 71	fvsnun2 7055	. . . . . . . . . . . 12 ⊢ ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) = (𝑎‘𝐽))
73	61, 72	breq12d 5087	. . . . . . . . . . 11 ⊢ ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ 𝑧𝐸(𝑎‘𝐽)))
74		ifpfal 1074	. . . . . . . . . . . . 13 ⊢ (¬ 𝐽 = 𝑁 → (if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)) ↔ 𝑧𝐸(𝑎‘𝐽)))
75	74	bicomd 222	. . . . . . . . . . . 12 ⊢ (¬ 𝐽 = 𝑁 → (𝑧𝐸(𝑎‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽))))
76	75	adantr 481	. . . . . . . . . . 11 ⊢ ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → (𝑧𝐸(𝑎‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽))))
77	73, 76	bitrd 278	. . . . . . . . . 10 ⊢ ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽))))
78	60, 77	pm2.61ian 809	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽))))
79	78	adantl 482	. . . . . . . 8 ⊢ ((𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽))))
80		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝐼 = 𝑁 → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁))
81	80	breq1d 5084	. . . . . . . . . 10 ⊢ (𝐼 = 𝑁 → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽)))
82		ifptru 1073	. . . . . . . . . 10 ⊢ (𝐼 = 𝑁 → (if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽))) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽))))
83	81, 82	bibi12d 346	. . . . . . . . 9 ⊢ (𝐼 = 𝑁 → (((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))) ↔ ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)))))
84	83	adantr 481	. . . . . . . 8 ⊢ ((𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → (((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))) ↔ ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)))))
85	79, 84	mpbird 256	. . . . . . 7 ⊢ ((𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))))
86	62	adantl 482	. . . . . . . . . . . . 13 ⊢ ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → 𝑁 ∈ ω)
87	64	adantl 482	. . . . . . . . . . . . 13 ⊢ ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → 𝑧 ∈ 𝑀)
88		neqne 2951	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐼 = 𝑁 → 𝐼 ≠ 𝑁)
89		simpll2 1212	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝐼 ∈ ω)
90	89	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)) → 𝐼 ∈ ω)
91	88, 90	anim12ci 614	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → (𝐼 ∈ ω ∧ 𝐼 ≠ 𝑁))
92		eldifsn 4720	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (ω ∖ {𝑁}) ↔ (𝐼 ∈ ω ∧ 𝐼 ≠ 𝑁))
93	91, 92	sylibr 233	. . . . . . . . . . . . 13 ⊢ ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → 𝐼 ∈ (ω ∖ {𝑁}))
94	86, 87, 50, 93	fvsnun2 7055	. . . . . . . . . . . 12 ⊢ ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼) = (𝑎‘𝐼))
95	52	adantl 482	. . . . . . . . . . . 12 ⊢ ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) = 𝑧)
96	94, 95	breq12d 5087	. . . . . . . . . . 11 ⊢ ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ (𝑎‘𝐼)𝐸𝑧))
97	96	adantl 482	. . . . . . . . . 10 ⊢ ((𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ (𝑎‘𝐼)𝐸𝑧))
98	55	breq2d 5086	. . . . . . . . . . . 12 ⊢ (𝐽 = 𝑁 → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)))
99		ifptru 1073	. . . . . . . . . . . 12 ⊢ (𝐽 = 𝑁 → (if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)) ↔ (𝑎‘𝐼)𝐸𝑧))
100	98, 99	bibi12d 346	. . . . . . . . . . 11 ⊢ (𝐽 = 𝑁 → (((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽))) ↔ ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ (𝑎‘𝐼)𝐸𝑧)))
101	100	adantr 481	. . . . . . . . . 10 ⊢ ((𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)))) → (((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽))) ↔ ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ (𝑎‘𝐼)𝐸𝑧)))
102	97, 101	mpbird 256	. . . . . . . . 9 ⊢ ((𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽))))
103	94	adantl 482	. . . . . . . . . . 11 ⊢ ((¬ 𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)))) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼) = (𝑎‘𝐼))
104	72	adantrl 713	. . . . . . . . . . 11 ⊢ ((¬ 𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)))) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) = (𝑎‘𝐽))
105	103, 104	breq12d 5087	. . . . . . . . . 10 ⊢ ((¬ 𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ (𝑎‘𝐼)𝐸(𝑎‘𝐽)))
106		ifpfal 1074	. . . . . . . . . . . 12 ⊢ (¬ 𝐽 = 𝑁 → (if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)) ↔ (𝑎‘𝐼)𝐸(𝑎‘𝐽)))
107	106	bicomd 222	. . . . . . . . . . 11 ⊢ (¬ 𝐽 = 𝑁 → ((𝑎‘𝐼)𝐸(𝑎‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽))))
108	107	adantr 481	. . . . . . . . . 10 ⊢ ((¬ 𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)))) → ((𝑎‘𝐼)𝐸(𝑎‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽))))
109	105, 108	bitrd 278	. . . . . . . . 9 ⊢ ((¬ 𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽))))
110	102, 109	pm2.61ian 809	. . . . . . . 8 ⊢ ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽))))
111		ifpfal 1074	. . . . . . . . . 10 ⊢ (¬ 𝐼 = 𝑁 → (if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽))) ↔ if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽))))
112	111	bicomd 222	. . . . . . . . 9 ⊢ (¬ 𝐼 = 𝑁 → (if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))))
113	112	adantr 481	. . . . . . . 8 ⊢ ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → (if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))))
114	110, 113	bitrd 278	. . . . . . 7 ⊢ ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))))
115	85, 114	pm2.61ian 809	. . . . . 6 ⊢ (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))))
116	48, 115	bitrd 278	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω)) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω) ∧ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽)) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))))
117	45, 116	mpdan 684	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω) ∧ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽)) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))))
118	5, 117	bitrd 278	. . 3 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀 ↑m ω) ∣ (𝑏‘𝐼)𝐸(𝑏‘𝐽)} ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))))
119	118	ralbidva 3111	. 2 ⊢ (((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) → (∀𝑧 ∈ 𝑀 ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀 ↑m ω) ∣ (𝑏‘𝐼)𝐸(𝑏‘𝐽)} ↔ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))))
120	119	rabbidva 3413	1 ⊢ ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀 ↑m ω) ∣ (𝑏‘𝐼)𝐸(𝑏‘𝐽)}} = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  if-wif 1060   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  {crab 3068  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886  ∅c0 4256  {csn 4561  ⟨cop 4567   class class class wbr 5074   ↾ cres 5591  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ωcom 7712   ↑_m cmap 8615

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-map 8617

This theorem is referenced by:  satfv1  33325