Theorem satfv1lem 36213

Description: Lemma for satfv1 36214. (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 6884	. . . . . . 7 ⊢ (𝑏 = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) → (𝑏‘𝐼) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼))
2		fveq1 6884	. . . . . . 7 ⊢ (𝑏 = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) → (𝑏‘𝐽) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽))
3	1, 2	breq12d 5115	. . . . . 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 3644	. . . . 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 3471	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → 𝑁 ∈ V)
7	6	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → 𝑁 ∈ V)
8	7	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝑁 ∈ V)
9		simpr 490	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝑧 ∈ 𝑀)
10	8, 9	fsnd 6869	. . . . . . . 8 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → {⟨𝑁, 𝑧⟩}:{𝑁}⟶𝑀)
11		elmapex 8871	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (𝑀 ∈ V ∧ ω ∈ V))
12	11	simpld 500	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → 𝑀 ∈ V)
13	12	adantr 486	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (𝑀 ↑m ω) ∧ 𝑧 ∈ 𝑀) → 𝑀 ∈ V)
14		snex 5396	. . . . . . . . . . 11 ⊢ {𝑁} ∈ V
15	14	a1i 11	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (𝑀 ↑m ω) ∧ 𝑧 ∈ 𝑀) → {𝑁} ∈ V)
16	13, 15	elmapd 8863	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝑀 ↑m ω) ∧ 𝑧 ∈ 𝑀) → ({⟨𝑁, 𝑧⟩} ∈ (𝑀 ↑m {𝑁}) ↔ {⟨𝑁, 𝑧⟩}:{𝑁}⟶𝑀))
17	16	adantll 727	. . . . . . . 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 260	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → {⟨𝑁, 𝑧⟩} ∈ (𝑀 ↑m {𝑁}))
19		elmapi 8872	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → 𝑎:ω⟶𝑀)
20		difssd 4073	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (ω ∖ {𝑁}) ⊆ ω)
21	19, 20	fssresd 6749	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (𝑎 ↾ (ω ∖ {𝑁})):(ω ∖ {𝑁})⟶𝑀)
22		omex 9647	. . . . . . . . . . . . 13 ⊢ ω ∈ V
23	22	difexi 5291	. . . . . . . . . . . 12 ⊢ (ω ∖ {𝑁}) ∈ V
24	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (ω ∖ {𝑁}) ∈ V)
25	12, 24	elmapd 8863	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → ((𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀 ↑m (ω ∖ {𝑁})) ↔ (𝑎 ↾ (ω ∖ {𝑁})):(ω ∖ {𝑁})⟶𝑀))
26	21, 25	mpbird 260	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀 ↑m (ω ∖ {𝑁})))
27	26	adantl 487	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) → (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀 ↑m (ω ∖ {𝑁})))
28	27	adantr 486	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀 ↑m (ω ∖ {𝑁})))
29		res0 5973	. . . . . . . . . 10 ⊢ ({⟨𝑁, 𝑧⟩} ↾ ∅) = ∅
30		res0 5973	. . . . . . . . . 10 ⊢ ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ∅) = ∅
31	29, 30	eqtr4i 2786	. . . . . . . . 9 ⊢ ({⟨𝑁, 𝑧⟩} ↾ ∅) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ∅)
32		disjdif 4425	. . . . . . . . . 10 ⊢ ({𝑁} ∩ (ω ∖ {𝑁})) = ∅
33	32	reseq2i 5966	. . . . . . . . 9 ⊢ ({⟨𝑁, 𝑧⟩} ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ({⟨𝑁, 𝑧⟩} ↾ ∅)
34	32	reseq2i 5966	. . . . . . . . 9 ⊢ ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ∅)
35	31, 33, 34	3eqtr4i 2793	. . . . . . . 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 8911	. . . . . . 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 1398	. . . . . 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 4157	. . . . . . . . . 10 ⊢ ({𝑁} ∪ (ω ∖ {𝑁})) = ((ω ∖ {𝑁}) ∪ {𝑁})
40		difsnid 4767	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → ((ω ∖ {𝑁}) ∪ {𝑁}) = ω)
41	39, 40	eqtr2id 2808	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → ω = ({𝑁} ∪ (ω ∖ {𝑁})))
42	41	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ω = ({𝑁} ∪ (ω ∖ {𝑁})))
43	42	ad2antrr 739	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → ω = ({𝑁} ∪ (ω ∖ {𝑁})))
44	43	oveq2d 7436	. . . . . 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 2863	. . . . 5 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))
46		ibar 538	. . . . . . . 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 487	. . . . . . 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 226	. . . . . 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 1231	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝑁 ∈ ω)
50		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))
51	49, 9, 50	fvsnun1 7187	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) = 𝑧)
52	51	adantr 486	. . . . . . . . . . . . 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 5115	. . . . . . . . . . . 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 487	. . . . . . . . . . 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 6885	. . . . . . . . . . . . . 14 ⊢ (𝐽 = 𝑁 → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁))
56	55	breq2d 5114	. . . . . . . . . . . . 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 1091	. . . . . . . . . . . . 13 ⊢ (𝐽 = 𝑁 → (if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)) ↔ 𝑧𝐸𝑧))
58	56, 57	bibi12d 348	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 260	. . . . . . . . . 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 487	. . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 487	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 487	. . . . . . . . . . . . 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 2963	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐽 = 𝑁 → 𝐽 ≠ 𝑁)
67		simpll3 1233	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝐽 ∈ ω)
68	67	adantr 486	. . . . . . . . . . . . . . 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 626	. . . . . . . . . . . . . 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 4744	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (ω ∖ {𝑁}) ↔ (𝐽 ∈ ω ∧ 𝐽 ≠ 𝑁))
71	69, 70	sylibr 237	. . . . . . . . . . . . 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 7188	. . . . . . . . . . . 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 5115	. . . . . . . . . . 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 1092	. . . . . . . . . . . . 13 ⊢ (¬ 𝐽 = 𝑁 → (if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)) ↔ 𝑧𝐸(𝑎‘𝐽)))
75	74	bicomd 226	. . . . . . . . . . . 12 ⊢ (¬ 𝐽 = 𝑁 → (𝑧𝐸(𝑎‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽))))
76	75	adantr 486	. . . . . . . . . . 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 282	. . . . . . . . . 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 824	. . . . . . . . 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 487	. . . . . . . 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 6885	. . . . . . . . . . 11 ⊢ (𝐼 = 𝑁 → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁))
81	80	breq1d 5112	. . . . . . . . . 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 1091	. . . . . . . . . 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 348	. . . . . . . . 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 486	. . . . . . . 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 260	. . . . . . 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 487	. . . . . . . . . . . . 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 487	. . . . . . . . . . . . 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 2963	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐼 = 𝑁 → 𝐼 ≠ 𝑁)
89		simpll2 1232	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝐼 ∈ ω)
90	89	adantr 486	. . . . . . . . . . . . . . 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 626	. . . . . . . . . . . . . 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 4744	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (ω ∖ {𝑁}) ↔ (𝐼 ∈ ω ∧ 𝐼 ≠ 𝑁))
93	91, 92	sylibr 237	. . . . . . . . . . . . 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 7188	. . . . . . . . . . . 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 487	. . . . . . . . . . . 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 5115	. . . . . . . . . . 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 487	. . . . . . . . . 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 5114	. . . . . . . . . . . 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 1091	. . . . . . . . . . . 12 ⊢ (𝐽 = 𝑁 → (if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)) ↔ (𝑎‘𝐼)𝐸𝑧))
100	98, 99	bibi12d 348	. . . . . . . . . . 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 486	. . . . . . . . . 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 260	. . . . . . . . 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 487	. . . . . . . . . . 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 729	. . . . . . . . . . 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 5115	. . . . . . . . . 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 1092	. . . . . . . . . . . 12 ⊢ (¬ 𝐽 = 𝑁 → (if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)) ↔ (𝑎‘𝐼)𝐸(𝑎‘𝐽)))
107	106	bicomd 226	. . . . . . . . . . 11 ⊢ (¬ 𝐽 = 𝑁 → ((𝑎‘𝐼)𝐸(𝑎‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽))))
108	107	adantr 486	. . . . . . . . . 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 282	. . . . . . . . 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 824	. . . . . . . 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 1092	. . . . . . . . . 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 226	. . . . . . . . 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 486	. . . . . . . 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 282	. . . . . . 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 824	. . . . . 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 282	. . . . 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 700	. . . 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 282	. . 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 3183	. 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 3418	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)))})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  if-wif 1078   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  {crab 3412  Vcvv 3450   ∖ cdif 4045   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  {csn 4584  ⟨cop 4834   class class class wbr 5102   ↾ cres 5652  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  ωcom 7877   ↑_m cmap 8850

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-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9645

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  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-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-map 8852

This theorem is used by:  satfv1  36214