Theorem satfv1lem 32609

Description: Lemma for satfv1 32610. (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 6669	. . . . . . 7 ⊢ (𝑏 = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) → (𝑏‘𝐼) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼))
2		fveq1 6669	. . . . . . 7 ⊢ (𝑏 = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) → (𝑏‘𝐽) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽))
3	1, 2	breq12d 5079	. . . . . 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 3680	. . . . 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 3512	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → 𝑁 ∈ V)
7	6	3ad2ant1 1129	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → 𝑁 ∈ V)
8	7	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝑁 ∈ V)
9		simpr 487	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝑧 ∈ 𝑀)
10	8, 9	fsnd 6657	. . . . . . . 8 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → {⟨𝑁, 𝑧⟩}:{𝑁}⟶𝑀)
11		elmapex 8427	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (𝑀 ∈ V ∧ ω ∈ V))
12	11	simpld 497	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → 𝑀 ∈ V)
13	12	adantr 483	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (𝑀 ↑m ω) ∧ 𝑧 ∈ 𝑀) → 𝑀 ∈ V)
14		snex 5332	. . . . . . . . . . 11 ⊢ {𝑁} ∈ V
15	14	a1i 11	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (𝑀 ↑m ω) ∧ 𝑧 ∈ 𝑀) → {𝑁} ∈ V)
16	13, 15	elmapd 8420	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝑀 ↑m ω) ∧ 𝑧 ∈ 𝑀) → ({⟨𝑁, 𝑧⟩} ∈ (𝑀 ↑m {𝑁}) ↔ {⟨𝑁, 𝑧⟩}:{𝑁}⟶𝑀))
17	16	adantll 712	. . . . . . . 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 259	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → {⟨𝑁, 𝑧⟩} ∈ (𝑀 ↑m {𝑁}))
19		elmapi 8428	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → 𝑎:ω⟶𝑀)
20		difssd 4109	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (ω ∖ {𝑁}) ⊆ ω)
21	19, 20	fssresd 6545	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (𝑎 ↾ (ω ∖ {𝑁})):(ω ∖ {𝑁})⟶𝑀)
22		omex 9106	. . . . . . . . . . . . 13 ⊢ ω ∈ V
23	22	difexi 5232	. . . . . . . . . . . 12 ⊢ (ω ∖ {𝑁}) ∈ V
24	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (ω ∖ {𝑁}) ∈ V)
25	12, 24	elmapd 8420	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → ((𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀 ↑m (ω ∖ {𝑁})) ↔ (𝑎 ↾ (ω ∖ {𝑁})):(ω ∖ {𝑁})⟶𝑀))
26	21, 25	mpbird 259	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑀 ↑m ω) → (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀 ↑m (ω ∖ {𝑁})))
27	26	adantl 484	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) → (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀 ↑m (ω ∖ {𝑁})))
28	27	adantr 483	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀 ↑m (ω ∖ {𝑁})))
29		res0 5857	. . . . . . . . . 10 ⊢ ({⟨𝑁, 𝑧⟩} ↾ ∅) = ∅
30		res0 5857	. . . . . . . . . 10 ⊢ ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ∅) = ∅
31	29, 30	eqtr4i 2847	. . . . . . . . 9 ⊢ ({⟨𝑁, 𝑧⟩} ↾ ∅) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ∅)
32		disjdif 4421	. . . . . . . . . 10 ⊢ ({𝑁} ∩ (ω ∖ {𝑁})) = ∅
33	32	reseq2i 5850	. . . . . . . . 9 ⊢ ({⟨𝑁, 𝑧⟩} ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ({⟨𝑁, 𝑧⟩} ↾ ∅)
34	32	reseq2i 5850	. . . . . . . . 9 ⊢ ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ∅)
35	31, 33, 34	3eqtr4i 2854	. . . . . . . 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 8444	. . . . . . 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 1367	. . . . . 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 4129	. . . . . . . . . 10 ⊢ ({𝑁} ∪ (ω ∖ {𝑁})) = ((ω ∖ {𝑁}) ∪ {𝑁})
40		difsnid 4743	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → ((ω ∖ {𝑁}) ∪ {𝑁}) = ω)
41	39, 40	syl5req 2869	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → ω = ({𝑁} ∪ (ω ∖ {𝑁})))
42	41	3ad2ant1 1129	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ω = ({𝑁} ∪ (ω ∖ {𝑁})))
43	42	ad2antrr 724	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → ω = ({𝑁} ∪ (ω ∖ {𝑁})))
44	43	oveq2d 7172	. . . . . 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 2916	. . . . 5 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀 ↑m ω))
46		ibar 531	. . . . . . . 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 484	. . . . . . 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 225	. . . . . 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 1208	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝑁 ∈ ω)
50		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))
51	49, 9, 50	fvsnun1 6944	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) = 𝑧)
52	51	adantr 483	. . . . . . . . . . . . 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 5079	. . . . . . . . . . . 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 484	. . . . . . . . . . 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 6670	. . . . . . . . . . . . . 14 ⊢ (𝐽 = 𝑁 → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁))
56	55	breq2d 5078	. . . . . . . . . . . . 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 1068	. . . . . . . . . . . . 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 483	. . . . . . . . . . 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 259	. . . . . . . . . 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 484	. . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 3024	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐽 = 𝑁 → 𝐽 ≠ 𝑁)
67		simpll3 1210	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝐽 ∈ ω)
68	67	adantr 483	. . . . . . . . . . . . . . 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 615	. . . . . . . . . . . . . 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 4719	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (ω ∖ {𝑁}) ↔ (𝐽 ∈ ω ∧ 𝐽 ≠ 𝑁))
71	69, 70	sylibr 236	. . . . . . . . . . . . 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 6945	. . . . . . . . . . . 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 5079	. . . . . . . . . . 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 1069	. . . . . . . . . . . . 13 ⊢ (¬ 𝐽 = 𝑁 → (if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)) ↔ 𝑧𝐸(𝑎‘𝐽)))
75	74	bicomd 225	. . . . . . . . . . . 12 ⊢ (¬ 𝐽 = 𝑁 → (𝑧𝐸(𝑎‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽))))
76	75	adantr 483	. . . . . . . . . . 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 281	. . . . . . . . . 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 810	. . . . . . . . 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 484	. . . . . . . 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 6670	. . . . . . . . . . 11 ⊢ (𝐼 = 𝑁 → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁))
81	80	breq1d 5076	. . . . . . . . . 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 1068	. . . . . . . . . 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 483	. . . . . . . 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 259	. . . . . . 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 484	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 3024	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐼 = 𝑁 → 𝐼 ≠ 𝑁)
89		simpll2 1209	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀 ↑m ω)) ∧ 𝑧 ∈ 𝑀) → 𝐼 ∈ ω)
90	89	adantr 483	. . . . . . . . . . . . . . 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 615	. . . . . . . . . . . . . 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 4719	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (ω ∖ {𝑁}) ↔ (𝐼 ∈ ω ∧ 𝐼 ≠ 𝑁))
93	91, 92	sylibr 236	. . . . . . . . . . . . 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 6945	. . . . . . . . . . . 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 484	. . . . . . . . . . . 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 5079	. . . . . . . . . . 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 484	. . . . . . . . . 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 5078	. . . . . . . . . . . 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 1068	. . . . . . . . . . . 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 483	. . . . . . . . . 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 259	. . . . . . . . 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 484	. . . . . . . . . . 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 714	. . . . . . . . . . 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 5079	. . . . . . . . . 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 1069	. . . . . . . . . . . 12 ⊢ (¬ 𝐽 = 𝑁 → (if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)) ↔ (𝑎‘𝐼)𝐸(𝑎‘𝐽)))
107	106	bicomd 225	. . . . . . . . . . 11 ⊢ (¬ 𝐽 = 𝑁 → ((𝑎‘𝐼)𝐸(𝑎‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽))))
108	107	adantr 483	. . . . . . . . . 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 281	. . . . . . . . 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 810	. . . . . . . 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 1069	. . . . . . . . . 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 225	. . . . . . . . 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 483	. . . . . . . 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 281	. . . . . . 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 810	. . . . . 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 281	. . . . 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 685	. . . 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 281	. . 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 3196	. 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 3478	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 208   ∧ wa 398  if-wif 1057   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  {crab 3142  Vcvv 3494   ∖ cdif 3933   ∪ cun 3934   ∩ cin 3935  ∅c0 4291  {csn 4567  ⟨cop 4573   class class class wbr 5066   ↾ cres 5557  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  ωcom 7580   ↑_m cmap 8406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-map 8408

This theorem is referenced by:  satfv1  32610