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Theorem satfv1lem 35346
Description: Lemma for satfv1 35347. (Contributed by AV, 9-Nov-2023.)
Assertion
Ref Expression
satfv1lem ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀m ω) ∣ (𝑏𝐼)𝐸(𝑏𝐽)}} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))})
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
StepHypRef Expression
1 fveq1 6905 . . . . . . 7 (𝑏 = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) → (𝑏𝐼) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼))
2 fveq1 6905 . . . . . . 7 (𝑏 = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) → (𝑏𝐽) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽))
31, 2breq12d 5160 . . . . . 6 (𝑏 = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) → ((𝑏𝐼)𝐸(𝑏𝐽) ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽)))
43elrab 3694 . . . . 5 (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀m ω) ∣ (𝑏𝐼)𝐸(𝑏𝐽)} ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω) ∧ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽)))
54a1i 11 . . . 4 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀m ω) ∣ (𝑏𝐼)𝐸(𝑏𝐽)} ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω) ∧ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽))))
6 elex 3498 . . . . . . . . . . 11 (𝑁 ∈ ω → 𝑁 ∈ V)
763ad2ant1 1132 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → 𝑁 ∈ V)
87ad2antrr 726 . . . . . . . . 9 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → 𝑁 ∈ V)
9 simpr 484 . . . . . . . . 9 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → 𝑧𝑀)
108, 9fsnd 6891 . . . . . . . 8 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → {⟨𝑁, 𝑧⟩}:{𝑁}⟶𝑀)
11 elmapex 8886 . . . . . . . . . . . 12 (𝑎 ∈ (𝑀m ω) → (𝑀 ∈ V ∧ ω ∈ V))
1211simpld 494 . . . . . . . . . . 11 (𝑎 ∈ (𝑀m ω) → 𝑀 ∈ V)
1312adantr 480 . . . . . . . . . 10 ((𝑎 ∈ (𝑀m ω) ∧ 𝑧𝑀) → 𝑀 ∈ V)
14 snex 5441 . . . . . . . . . . 11 {𝑁} ∈ V
1514a1i 11 . . . . . . . . . 10 ((𝑎 ∈ (𝑀m ω) ∧ 𝑧𝑀) → {𝑁} ∈ V)
1613, 15elmapd 8878 . . . . . . . . 9 ((𝑎 ∈ (𝑀m ω) ∧ 𝑧𝑀) → ({⟨𝑁, 𝑧⟩} ∈ (𝑀m {𝑁}) ↔ {⟨𝑁, 𝑧⟩}:{𝑁}⟶𝑀))
1716adantll 714 . . . . . . . 8 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → ({⟨𝑁, 𝑧⟩} ∈ (𝑀m {𝑁}) ↔ {⟨𝑁, 𝑧⟩}:{𝑁}⟶𝑀))
1810, 17mpbird 257 . . . . . . 7 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → {⟨𝑁, 𝑧⟩} ∈ (𝑀m {𝑁}))
19 elmapi 8887 . . . . . . . . . . 11 (𝑎 ∈ (𝑀m ω) → 𝑎:ω⟶𝑀)
20 difssd 4146 . . . . . . . . . . 11 (𝑎 ∈ (𝑀m ω) → (ω ∖ {𝑁}) ⊆ ω)
2119, 20fssresd 6775 . . . . . . . . . 10 (𝑎 ∈ (𝑀m ω) → (𝑎 ↾ (ω ∖ {𝑁})):(ω ∖ {𝑁})⟶𝑀)
22 omex 9680 . . . . . . . . . . . . 13 ω ∈ V
2322difexi 5335 . . . . . . . . . . . 12 (ω ∖ {𝑁}) ∈ V
2423a1i 11 . . . . . . . . . . 11 (𝑎 ∈ (𝑀m ω) → (ω ∖ {𝑁}) ∈ V)
2512, 24elmapd 8878 . . . . . . . . . 10 (𝑎 ∈ (𝑀m ω) → ((𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀m (ω ∖ {𝑁})) ↔ (𝑎 ↾ (ω ∖ {𝑁})):(ω ∖ {𝑁})⟶𝑀))
2621, 25mpbird 257 . . . . . . . . 9 (𝑎 ∈ (𝑀m ω) → (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀m (ω ∖ {𝑁})))
2726adantl 481 . . . . . . . 8 (((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) → (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀m (ω ∖ {𝑁})))
2827adantr 480 . . . . . . 7 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀m (ω ∖ {𝑁})))
29 res0 6003 . . . . . . . . . 10 ({⟨𝑁, 𝑧⟩} ↾ ∅) = ∅
30 res0 6003 . . . . . . . . . 10 ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ∅) = ∅
3129, 30eqtr4i 2765 . . . . . . . . 9 ({⟨𝑁, 𝑧⟩} ↾ ∅) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ∅)
32 disjdif 4477 . . . . . . . . . 10 ({𝑁} ∩ (ω ∖ {𝑁})) = ∅
3332reseq2i 5996 . . . . . . . . 9 ({⟨𝑁, 𝑧⟩} ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ({⟨𝑁, 𝑧⟩} ↾ ∅)
3432reseq2i 5996 . . . . . . . . 9 ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ∅)
3531, 33, 343eqtr4i 2772 . . . . . . . 8 ({⟨𝑁, 𝑧⟩} ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ({𝑁} ∩ (ω ∖ {𝑁})))
3635a1i 11 . . . . . . 7 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → ({⟨𝑁, 𝑧⟩} ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ({𝑁} ∩ (ω ∖ {𝑁}))))
37 elmapresaun 8918 . . . . . . 7 (({⟨𝑁, 𝑧⟩} ∈ (𝑀m {𝑁}) ∧ (𝑎 ↾ (ω ∖ {𝑁})) ∈ (𝑀m (ω ∖ {𝑁})) ∧ ({⟨𝑁, 𝑧⟩} ↾ ({𝑁} ∩ (ω ∖ {𝑁}))) = ((𝑎 ↾ (ω ∖ {𝑁})) ↾ ({𝑁} ∩ (ω ∖ {𝑁})))) → ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ({𝑁} ∪ (ω ∖ {𝑁}))))
3818, 28, 36, 37syl3anc 1370 . . . . . 6 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ({𝑁} ∪ (ω ∖ {𝑁}))))
39 uncom 4167 . . . . . . . . . 10 ({𝑁} ∪ (ω ∖ {𝑁})) = ((ω ∖ {𝑁}) ∪ {𝑁})
40 difsnid 4814 . . . . . . . . . 10 (𝑁 ∈ ω → ((ω ∖ {𝑁}) ∪ {𝑁}) = ω)
4139, 40eqtr2id 2787 . . . . . . . . 9 (𝑁 ∈ ω → ω = ({𝑁} ∪ (ω ∖ {𝑁})))
42413ad2ant1 1132 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ω = ({𝑁} ∪ (ω ∖ {𝑁})))
4342ad2antrr 726 . . . . . . 7 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → ω = ({𝑁} ∪ (ω ∖ {𝑁})))
4443oveq2d 7446 . . . . . 6 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → (𝑀m ω) = (𝑀m ({𝑁} ∪ (ω ∖ {𝑁}))))
4538, 44eleqtrrd 2841 . . . . 5 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))
46 ibar 528 . . . . . . . 8 (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω) ∧ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽))))
4746adantl 481 . . . . . . 7 (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω) ∧ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽))))
4847bicomd 223 . . . . . 6 (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m 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 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → 𝑁 ∈ ω)
50 eqid 2734 . . . . . . . . . . . . . . 15 ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) = ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))
5149, 9, 50fvsnun1 7201 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) = 𝑧)
5251adantr 480 . . . . . . . . . . . . 13 (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) = 𝑧)
5352, 52breq12d 5160 . . . . . . . . . . . 12 (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ 𝑧𝐸𝑧))
5453adantl 481 . . . . . . . . . . 11 ((𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ 𝑧𝐸𝑧))
55 fveq2 6906 . . . . . . . . . . . . . 14 (𝐽 = 𝑁 → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁))
5655breq2d 5159 . . . . . . . . . . . . 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 1074 . . . . . . . . . . . . 13 (𝐽 = 𝑁 → (if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)) ↔ 𝑧𝐸𝑧))
5856, 57bibi12d 345 . . . . . . . . . . . 12 (𝐽 = 𝑁 → (((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽))) ↔ ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ 𝑧𝐸𝑧)))
5958adantr 480 . . . . . . . . . . 11 ((𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → (((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽))) ↔ ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ 𝑧𝐸𝑧)))
6054, 59mpbird 257 . . . . . . . . . 10 ((𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽))))
6152adantl 481 . . . . . . . . . . . 12 ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) = 𝑧)
6249adantr 480 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)) → 𝑁 ∈ ω)
6362adantl 481 . . . . . . . . . . . . 13 ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → 𝑁 ∈ ω)
649adantr 480 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)) → 𝑧𝑀)
6564adantl 481 . . . . . . . . . . . . 13 ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → 𝑧𝑀)
66 neqne 2945 . . . . . . . . . . . . . . 15 𝐽 = 𝑁𝐽𝑁)
67 simpll3 1213 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → 𝐽 ∈ ω)
6867adantr 480 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)) → 𝐽 ∈ ω)
6966, 68anim12ci 614 . . . . . . . . . . . . . 14 ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → (𝐽 ∈ ω ∧ 𝐽𝑁))
70 eldifsn 4790 . . . . . . . . . . . . . 14 (𝐽 ∈ (ω ∖ {𝑁}) ↔ (𝐽 ∈ ω ∧ 𝐽𝑁))
7169, 70sylibr 234 . . . . . . . . . . . . 13 ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → 𝐽 ∈ (ω ∖ {𝑁}))
7263, 65, 50, 71fvsnun2 7202 . . . . . . . . . . . 12 ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) = (𝑎𝐽))
7361, 72breq12d 5160 . . . . . . . . . . 11 ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ 𝑧𝐸(𝑎𝐽)))
74 ifpfal 1075 . . . . . . . . . . . . 13 𝐽 = 𝑁 → (if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)) ↔ 𝑧𝐸(𝑎𝐽)))
7574bicomd 223 . . . . . . . . . . . 12 𝐽 = 𝑁 → (𝑧𝐸(𝑎𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽))))
7675adantr 480 . . . . . . . . . . 11 ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → (𝑧𝐸(𝑎𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽))))
7773, 76bitrd 279 . . . . . . . . . 10 ((¬ 𝐽 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽))))
7860, 77pm2.61ian 812 . . . . . . . . 9 (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽))))
7978adantl 481 . . . . . . . 8 ((𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽))))
80 fveq2 6906 . . . . . . . . . . 11 (𝐼 = 𝑁 → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼) = (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁))
8180breq1d 5157 . . . . . . . . . 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 1074 . . . . . . . . . 10 (𝐼 = 𝑁 → (if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽))) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽))))
8381, 82bibi12d 345 . . . . . . . . 9 (𝐼 = 𝑁 → (((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))) ↔ ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)))))
8483adantr 480 . . . . . . . 8 ((𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → (((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))) ↔ ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)))))
8579, 84mpbird 257 . . . . . . 7 ((𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))))
8662adantl 481 . . . . . . . . . . . . 13 ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → 𝑁 ∈ ω)
8764adantl 481 . . . . . . . . . . . . 13 ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → 𝑧𝑀)
88 neqne 2945 . . . . . . . . . . . . . . 15 𝐼 = 𝑁𝐼𝑁)
89 simpll2 1212 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → 𝐼 ∈ ω)
9089adantr 480 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)) → 𝐼 ∈ ω)
9188, 90anim12ci 614 . . . . . . . . . . . . . 14 ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → (𝐼 ∈ ω ∧ 𝐼𝑁))
92 eldifsn 4790 . . . . . . . . . . . . . 14 (𝐼 ∈ (ω ∖ {𝑁}) ↔ (𝐼 ∈ ω ∧ 𝐼𝑁))
9391, 92sylibr 234 . . . . . . . . . . . . 13 ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → 𝐼 ∈ (ω ∖ {𝑁}))
9486, 87, 50, 93fvsnun2 7202 . . . . . . . . . . . 12 ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼) = (𝑎𝐼))
9552adantl 481 . . . . . . . . . . . 12 ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) = 𝑧)
9694, 95breq12d 5160 . . . . . . . . . . 11 ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ (𝑎𝐼)𝐸𝑧))
9796adantl 481 . . . . . . . . . 10 ((𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ (𝑎𝐼)𝐸𝑧))
9855breq2d 5159 . . . . . . . . . . . 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 1074 . . . . . . . . . . . 12 (𝐽 = 𝑁 → (if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)) ↔ (𝑎𝐼)𝐸𝑧))
10098, 99bibi12d 345 . . . . . . . . . . 11 (𝐽 = 𝑁 → (((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽))) ↔ ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ (𝑎𝐼)𝐸𝑧)))
101100adantr 480 . . . . . . . . . 10 ((𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)))) → (((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽))) ↔ ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝑁) ↔ (𝑎𝐼)𝐸𝑧)))
10297, 101mpbird 257 . . . . . . . . 9 ((𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽))))
10394adantl 481 . . . . . . . . . . 11 ((¬ 𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)))) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼) = (𝑎𝐼))
10472adantrl 716 . . . . . . . . . . 11 ((¬ 𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)))) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) = (𝑎𝐽))
105103, 104breq12d 5160 . . . . . . . . . 10 ((¬ 𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ (𝑎𝐼)𝐸(𝑎𝐽)))
106 ifpfal 1075 . . . . . . . . . . . 12 𝐽 = 𝑁 → (if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)) ↔ (𝑎𝐼)𝐸(𝑎𝐽)))
107106bicomd 223 . . . . . . . . . . 11 𝐽 = 𝑁 → ((𝑎𝐼)𝐸(𝑎𝐽) ↔ if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽))))
108107adantr 480 . . . . . . . . . 10 ((¬ 𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)))) → ((𝑎𝐼)𝐸(𝑎𝐽) ↔ if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽))))
109105, 108bitrd 279 . . . . . . . . 9 ((¬ 𝐽 = 𝑁 ∧ (¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽))))
110102, 109pm2.61ian 812 . . . . . . . 8 ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽))))
111 ifpfal 1075 . . . . . . . . . 10 𝐼 = 𝑁 → (if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽))) ↔ if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽))))
112111bicomd 223 . . . . . . . . 9 𝐼 = 𝑁 → (if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))))
113112adantr 480 . . . . . . . 8 ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → (if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))))
114110, 113bitrd 279 . . . . . . 7 ((¬ 𝐼 = 𝑁 ∧ ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω))) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))))
11585, 114pm2.61ian 812 . . . . . 6 (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))))
11648, 115bitrd 279 . . . . 5 (((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) ∧ ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω)) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω) ∧ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽)) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))))
11745, 116mpdan 687 . . . 4 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → ((({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ (𝑀m ω) ∧ (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐼)𝐸(({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁})))‘𝐽)) ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))))
1185, 117bitrd 279 . . 3 ((((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) ∧ 𝑧𝑀) → (({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀m ω) ∣ (𝑏𝐼)𝐸(𝑏𝐽)} ↔ if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))))
119118ralbidva 3173 . 2 (((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑎 ∈ (𝑀m ω)) → (∀𝑧𝑀 ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀m ω) ∣ (𝑏𝐼)𝐸(𝑏𝐽)} ↔ ∀𝑧𝑀 if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))))
120119rabbidva 3439 1 ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀m ω) ∣ (𝑏𝐼)𝐸(𝑏𝐽)}} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  if-wif 1062  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  {crab 3432  Vcvv 3477  cdif 3959  cun 3960  cin 3961  c0 4338  {csn 4630  cop 4636   class class class wbr 5147  cres 5690  wf 6558  cfv 6562  (class class class)co 7430  ωcom 7886  m cmap 8864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-map 8866
This theorem is referenced by:  satfv1  35347
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