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Theorem isf32lem2 9768
 Description: Lemma for isfin3-2 9781. Non-minimum implies that there is always another decrease. (Contributed by Stefan O'Rear, 5-Nov-2014.)
Hypotheses
Ref Expression
isf32lem.a (𝜑𝐹:ω⟶𝒫 𝐺)
isf32lem.b (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))
isf32lem.c (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)
Assertion
Ref Expression
isf32lem2 ((𝜑𝐴 ∈ ω) → ∃𝑎 ∈ ω (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎)))
Distinct variable groups:   𝑥,𝑎   𝐺,𝑎   𝜑,𝑎,𝑥   𝐴,𝑎,𝑥   𝐹,𝑎,𝑥
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem isf32lem2
Dummy variables 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isf32lem.c . . . . 5 (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)
21adantr 484 . . . 4 ((𝜑𝐴 ∈ ω) → ¬ ran 𝐹 ∈ ran 𝐹)
3 isf32lem.a . . . . . . . . . 10 (𝜑𝐹:ω⟶𝒫 𝐺)
43ffnd 6489 . . . . . . . . 9 (𝜑𝐹 Fn ω)
5 peano2 7585 . . . . . . . . 9 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
6 fnfvelrn 6826 . . . . . . . . 9 ((𝐹 Fn ω ∧ suc 𝐴 ∈ ω) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
74, 5, 6syl2an 598 . . . . . . . 8 ((𝜑𝐴 ∈ ω) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
87adantr 484 . . . . . . 7 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
9 intss1 4854 . . . . . . 7 ((𝐹‘suc 𝐴) ∈ ran 𝐹 ran 𝐹 ⊆ (𝐹‘suc 𝐴))
108, 9syl 17 . . . . . 6 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ran 𝐹 ⊆ (𝐹‘suc 𝐴))
11 fvelrnb 6702 . . . . . . . . . . 11 (𝐹 Fn ω → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹𝑐) = 𝑏))
124, 11syl 17 . . . . . . . . . 10 (𝜑 → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹𝑐) = 𝑏))
1312ad2antrr 725 . . . . . . . . 9 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹𝑐) = 𝑏))
14 simplrr 777 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → 𝑐 ∈ ω)
155ad3antlr 730 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → suc 𝐴 ∈ ω)
16 simpr 488 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → suc 𝐴𝑐)
17 simplrl 776 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
18 fveq2 6646 . . . . . . . . . . . . . . . . . . 19 (𝑏 = suc 𝐴 → (𝐹𝑏) = (𝐹‘suc 𝐴))
1918eqeq2d 2809 . . . . . . . . . . . . . . . . . 18 (𝑏 = suc 𝐴 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)))
2019imbi2d 344 . . . . . . . . . . . . . . . . 17 (𝑏 = suc 𝐴 → ((∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑏)) ↔ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴))))
21 fveq2 6646 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑑 → (𝐹𝑏) = (𝐹𝑑))
2221eqeq2d 2809 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑑 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹𝑑)))
2322imbi2d 344 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑑 → ((∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑏)) ↔ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑑))))
24 fveq2 6646 . . . . . . . . . . . . . . . . . . 19 (𝑏 = suc 𝑑 → (𝐹𝑏) = (𝐹‘suc 𝑑))
2524eqeq2d 2809 . . . . . . . . . . . . . . . . . 18 (𝑏 = suc 𝑑 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑)))
2625imbi2d 344 . . . . . . . . . . . . . . . . 17 (𝑏 = suc 𝑑 → ((∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑏)) ↔ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))))
27 fveq2 6646 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑐 → (𝐹𝑏) = (𝐹𝑐))
2827eqeq2d 2809 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑐 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹𝑐)))
2928imbi2d 344 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑐 → ((∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑏)) ↔ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑐))))
30 eqid 2798 . . . . . . . . . . . . . . . . . 18 (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)
31302a1i 12 . . . . . . . . . . . . . . . . 17 (suc 𝐴 ∈ ω → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)))
32 elex 3459 . . . . . . . . . . . . . . . . . . . . . . . 24 (suc 𝐴 ∈ ω → suc 𝐴 ∈ V)
33 sucexb 7507 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3432, 33sylibr 237 . . . . . . . . . . . . . . . . . . . . . . 23 (suc 𝐴 ∈ ω → 𝐴 ∈ V)
3534adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) → 𝐴 ∈ V)
36 sucssel 6252 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ V → (suc 𝐴𝑑𝐴𝑑))
3735, 36syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) → (suc 𝐴𝑑𝐴𝑑))
3837imp 410 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → 𝐴𝑑)
39 eleq2w 2873 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑑 → (𝐴𝑎𝐴𝑑))
40 suceq 6225 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑑 → suc 𝑎 = suc 𝑑)
4140fveq2d 6650 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑑 → (𝐹‘suc 𝑎) = (𝐹‘suc 𝑑))
42 fveq2 6646 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑑 → (𝐹𝑎) = (𝐹𝑑))
4341, 42eqeq12d 2814 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑑 → ((𝐹‘suc 𝑎) = (𝐹𝑎) ↔ (𝐹‘suc 𝑑) = (𝐹𝑑)))
4439, 43imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝑑 → ((𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ↔ (𝐴𝑑 → (𝐹‘suc 𝑑) = (𝐹𝑑))))
4544rspcv 3566 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ ω → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑑 → (𝐹‘suc 𝑑) = (𝐹𝑑))))
4645com23 86 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 ∈ ω → (𝐴𝑑 → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝑑) = (𝐹𝑑))))
4746ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → (𝐴𝑑 → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝑑) = (𝐹𝑑))))
4838, 47mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝑑) = (𝐹𝑑)))
49 eqtr3 2820 . . . . . . . . . . . . . . . . . . . 20 (((𝐹‘suc 𝐴) = (𝐹𝑑) ∧ (𝐹‘suc 𝑑) = (𝐹𝑑)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))
5049expcom 417 . . . . . . . . . . . . . . . . . . 19 ((𝐹‘suc 𝑑) = (𝐹𝑑) → ((𝐹‘suc 𝐴) = (𝐹𝑑) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑)))
5148, 50syl6 35 . . . . . . . . . . . . . . . . . 18 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → ((𝐹‘suc 𝐴) = (𝐹𝑑) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))))
5251a2d 29 . . . . . . . . . . . . . . . . 17 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → ((∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑑)) → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))))
5320, 23, 26, 29, 31, 52findsg 7593 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑐) → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑐)))
5453impr 458 . . . . . . . . . . . . . . 15 (((𝑐 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ (suc 𝐴𝑐 ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))) → (𝐹‘suc 𝐴) = (𝐹𝑐))
5514, 15, 16, 17, 54syl22anc 837 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → (𝐹‘suc 𝐴) = (𝐹𝑐))
56 eqimss 3971 . . . . . . . . . . . . . 14 ((𝐹‘suc 𝐴) = (𝐹𝑐) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
5755, 56syl 17 . . . . . . . . . . . . 13 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
585ad3antlr 730 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → suc 𝐴 ∈ ω)
59 simplrr 777 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝑐 ∈ ω)
60 simpr 488 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝑐 ⊆ suc 𝐴)
61 simplll 774 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝜑)
62 isf32lem.b . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))
633, 62, 1isf32lem1 9767 . . . . . . . . . . . . . 14 (((suc 𝐴 ∈ ω ∧ 𝑐 ∈ ω) ∧ (𝑐 ⊆ suc 𝐴𝜑)) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
6458, 59, 60, 61, 63syl22anc 837 . . . . . . . . . . . . 13 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
65 nnord 7571 . . . . . . . . . . . . . . . 16 (suc 𝐴 ∈ ω → Ord suc 𝐴)
665, 65syl 17 . . . . . . . . . . . . . . 15 (𝐴 ∈ ω → Ord suc 𝐴)
6766ad2antlr 726 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → Ord suc 𝐴)
68 nnord 7571 . . . . . . . . . . . . . . 15 (𝑐 ∈ ω → Ord 𝑐)
6968ad2antll 728 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → Ord 𝑐)
70 ordtri2or2 6256 . . . . . . . . . . . . . 14 ((Ord suc 𝐴 ∧ Ord 𝑐) → (suc 𝐴𝑐𝑐 ⊆ suc 𝐴))
7167, 69, 70syl2anc 587 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → (suc 𝐴𝑐𝑐 ⊆ suc 𝐴))
7257, 64, 71mpjaodan 956 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
7372anassrs 471 . . . . . . . . . . 11 ((((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) ∧ 𝑐 ∈ ω) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
74 sseq2 3941 . . . . . . . . . . 11 ((𝐹𝑐) = 𝑏 → ((𝐹‘suc 𝐴) ⊆ (𝐹𝑐) ↔ (𝐹‘suc 𝐴) ⊆ 𝑏))
7573, 74syl5ibcom 248 . . . . . . . . . 10 ((((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) ∧ 𝑐 ∈ ω) → ((𝐹𝑐) = 𝑏 → (𝐹‘suc 𝐴) ⊆ 𝑏))
7675rexlimdva 3243 . . . . . . . . 9 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (∃𝑐 ∈ ω (𝐹𝑐) = 𝑏 → (𝐹‘suc 𝐴) ⊆ 𝑏))
7713, 76sylbid 243 . . . . . . . 8 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝑏 ∈ ran 𝐹 → (𝐹‘suc 𝐴) ⊆ 𝑏))
7877ralrimiv 3148 . . . . . . 7 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ∀𝑏 ∈ ran 𝐹(𝐹‘suc 𝐴) ⊆ 𝑏)
79 ssint 4855 . . . . . . 7 ((𝐹‘suc 𝐴) ⊆ ran 𝐹 ↔ ∀𝑏 ∈ ran 𝐹(𝐹‘suc 𝐴) ⊆ 𝑏)
8078, 79sylibr 237 . . . . . 6 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝐹‘suc 𝐴) ⊆ ran 𝐹)
8110, 80eqssd 3932 . . . . 5 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ran 𝐹 = (𝐹‘suc 𝐴))
8281, 8eqeltrd 2890 . . . 4 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ran 𝐹 ∈ ran 𝐹)
832, 82mtand 815 . . 3 ((𝜑𝐴 ∈ ω) → ¬ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
84 rexnal 3201 . . 3 (∃𝑎 ∈ ω ¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ↔ ¬ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
8583, 84sylibr 237 . 2 ((𝜑𝐴 ∈ ω) → ∃𝑎 ∈ ω ¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
86 suceq 6225 . . . . . . . 8 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
8786fveq2d 6650 . . . . . . 7 (𝑥 = 𝑎 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑎))
88 fveq2 6646 . . . . . . 7 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
8987, 88sseq12d 3948 . . . . . 6 (𝑥 = 𝑎 → ((𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ↔ (𝐹‘suc 𝑎) ⊆ (𝐹𝑎)))
9089cbvralvw 3396 . . . . 5 (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ↔ ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎))
9162, 90sylib 221 . . . 4 (𝜑 → ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎))
9291adantr 484 . . 3 ((𝜑𝐴 ∈ ω) → ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎))
93 pm4.61 408 . . . . 5 (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ↔ (𝐴𝑎 ∧ ¬ (𝐹‘suc 𝑎) = (𝐹𝑎)))
94 dfpss2 4013 . . . . . . 7 ((𝐹‘suc 𝑎) ⊊ (𝐹𝑎) ↔ ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) ∧ ¬ (𝐹‘suc 𝑎) = (𝐹𝑎)))
9594simplbi2 504 . . . . . 6 ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → (¬ (𝐹‘suc 𝑎) = (𝐹𝑎) → (𝐹‘suc 𝑎) ⊊ (𝐹𝑎)))
9695anim2d 614 . . . . 5 ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → ((𝐴𝑎 ∧ ¬ (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
9793, 96syl5bi 245 . . . 4 ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
9897ralimi 3128 . . 3 (∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → ∀𝑎 ∈ ω (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
99 rexim 3204 . . 3 (∀𝑎 ∈ ω (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))) → (∃𝑎 ∈ ω ¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → ∃𝑎 ∈ ω (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
10092, 98, 993syl 18 . 2 ((𝜑𝐴 ∈ ω) → (∃𝑎 ∈ ω ¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → ∃𝑎 ∈ ω (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
10185, 100mpd 15 1 ((𝜑𝐴 ∈ ω) → ∃𝑎 ∈ ω (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881   ⊊ wpss 3882  𝒫 cpw 4497  ∩ cint 4839  ran crn 5521  Ord word 6159  suc csuc 6162   Fn wfn 6320  ⟶wf 6321  ‘cfv 6325  ωcom 7563 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-om 7564 This theorem is referenced by:  isf32lem5  9771
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