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Theorem isf32lem2 10423
Description: Lemma for isfin3-2 10436. 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 480 . . . 4 ((𝜑𝐴 ∈ ω) → ¬ ran 𝐹 ∈ ran 𝐹)
3 isf32lem.a . . . . . . . . . 10 (𝜑𝐹:ω⟶𝒫 𝐺)
43ffnd 6748 . . . . . . . . 9 (𝜑𝐹 Fn ω)
5 peano2 7929 . . . . . . . . 9 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
6 fnfvelrn 7114 . . . . . . . . 9 ((𝐹 Fn ω ∧ suc 𝐴 ∈ ω) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
74, 5, 6syl2an 595 . . . . . . . 8 ((𝜑𝐴 ∈ ω) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
87adantr 480 . . . . . . 7 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
9 intss1 4987 . . . . . . 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 6982 . . . . . . . . . . 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 484 . . . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . . . . . . 19 (𝑏 = suc 𝐴 → (𝐹𝑏) = (𝐹‘suc 𝐴))
1918eqeq2d 2751 . . . . . . . . . . . . . . . . . 18 (𝑏 = suc 𝐴 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)))
2019imbi2d 340 . . . . . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑑 → (𝐹𝑏) = (𝐹𝑑))
2221eqeq2d 2751 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑑 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹𝑑)))
2322imbi2d 340 . . . . . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . . . . . . 19 (𝑏 = suc 𝑑 → (𝐹𝑏) = (𝐹‘suc 𝑑))
2524eqeq2d 2751 . . . . . . . . . . . . . . . . . 18 (𝑏 = suc 𝑑 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑)))
2625imbi2d 340 . . . . . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑐 → (𝐹𝑏) = (𝐹𝑐))
2827eqeq2d 2751 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑐 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹𝑐)))
2928imbi2d 340 . . . . . . . . . . . . . . . . 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 2740 . . . . . . . . . . . . . . . . . 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 3509 . . . . . . . . . . . . . . . . . . . . . . . 24 (suc 𝐴 ∈ ω → suc 𝐴 ∈ V)
33 sucexb 7840 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3432, 33sylibr 234 . . . . . . . . . . . . . . . . . . . . . . 23 (suc 𝐴 ∈ ω → 𝐴 ∈ V)
3534adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) → 𝐴 ∈ V)
36 sucssel 6490 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ V → (suc 𝐴𝑑𝐴𝑑))
3735, 36syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) → (suc 𝐴𝑑𝐴𝑑))
3837imp 406 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → 𝐴𝑑)
39 eleq2w 2828 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑑 → (𝐴𝑎𝐴𝑑))
40 suceq 6461 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑑 → suc 𝑎 = suc 𝑑)
4140fveq2d 6924 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑑 → (𝐹‘suc 𝑎) = (𝐹‘suc 𝑑))
42 fveq2 6920 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑑 → (𝐹𝑎) = (𝐹𝑑))
4341, 42eqeq12d 2756 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑑 → ((𝐹‘suc 𝑎) = (𝐹𝑎) ↔ (𝐹‘suc 𝑑) = (𝐹𝑑)))
4439, 43imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝑑 → ((𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ↔ (𝐴𝑑 → (𝐹‘suc 𝑑) = (𝐹𝑑))))
4544rspcv 3631 . . . . . . . . . . . . . . . . . . . . . 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 2766 . . . . . . . . . . . . . . . . . . . 20 (((𝐹‘suc 𝐴) = (𝐹𝑑) ∧ (𝐹‘suc 𝑑) = (𝐹𝑑)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))
5049expcom 413 . . . . . . . . . . . . . . . . . . 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 7937 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑐) → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑐)))
5453impr 454 . . . . . . . . . . . . . . 15 (((𝑐 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ (suc 𝐴𝑐 ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))) → (𝐹‘suc 𝐴) = (𝐹𝑐))
5514, 15, 16, 17, 54syl22anc 838 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → (𝐹‘suc 𝐴) = (𝐹𝑐))
56 eqimss 4067 . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 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 10422 . . . . . . . . . . . . . 14 (((suc 𝐴 ∈ ω ∧ 𝑐 ∈ ω) ∧ (𝑐 ⊆ suc 𝐴𝜑)) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
6458, 59, 60, 61, 63syl22anc 838 . . . . . . . . . . . . 13 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
65 nnord 7911 . . . . . . . . . . . . . . . 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 7911 . . . . . . . . . . . . . . 15 (𝑐 ∈ ω → Ord 𝑐)
6968ad2antll 728 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → Ord 𝑐)
70 ordtri2or2 6494 . . . . . . . . . . . . . 14 ((Ord suc 𝐴 ∧ Ord 𝑐) → (suc 𝐴𝑐𝑐 ⊆ suc 𝐴))
7167, 69, 70syl2anc 583 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → (suc 𝐴𝑐𝑐 ⊆ suc 𝐴))
7257, 64, 71mpjaodan 959 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
7372anassrs 467 . . . . . . . . . . 11 ((((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) ∧ 𝑐 ∈ ω) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
74 sseq2 4035 . . . . . . . . . . 11 ((𝐹𝑐) = 𝑏 → ((𝐹‘suc 𝐴) ⊆ (𝐹𝑐) ↔ (𝐹‘suc 𝐴) ⊆ 𝑏))
7573, 74syl5ibcom 245 . . . . . . . . . 10 ((((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) ∧ 𝑐 ∈ ω) → ((𝐹𝑐) = 𝑏 → (𝐹‘suc 𝐴) ⊆ 𝑏))
7675rexlimdva 3161 . . . . . . . . 9 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (∃𝑐 ∈ ω (𝐹𝑐) = 𝑏 → (𝐹‘suc 𝐴) ⊆ 𝑏))
7713, 76sylbid 240 . . . . . . . 8 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝑏 ∈ ran 𝐹 → (𝐹‘suc 𝐴) ⊆ 𝑏))
7877ralrimiv 3151 . . . . . . 7 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ∀𝑏 ∈ ran 𝐹(𝐹‘suc 𝐴) ⊆ 𝑏)
79 ssint 4988 . . . . . . 7 ((𝐹‘suc 𝐴) ⊆ ran 𝐹 ↔ ∀𝑏 ∈ ran 𝐹(𝐹‘suc 𝐴) ⊆ 𝑏)
8078, 79sylibr 234 . . . . . 6 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝐹‘suc 𝐴) ⊆ ran 𝐹)
8110, 80eqssd 4026 . . . . 5 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ran 𝐹 = (𝐹‘suc 𝐴))
8281, 8eqeltrd 2844 . . . 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 3106 . . 3 (∃𝑎 ∈ ω ¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ↔ ¬ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
8583, 84sylibr 234 . 2 ((𝜑𝐴 ∈ ω) → ∃𝑎 ∈ ω ¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
86 suceq 6461 . . . . . . . 8 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
8786fveq2d 6924 . . . . . . 7 (𝑥 = 𝑎 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑎))
88 fveq2 6920 . . . . . . 7 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
8987, 88sseq12d 4042 . . . . . 6 (𝑥 = 𝑎 → ((𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ↔ (𝐹‘suc 𝑎) ⊆ (𝐹𝑎)))
9089cbvralvw 3243 . . . . 5 (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ↔ ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎))
9162, 90sylib 218 . . . 4 (𝜑 → ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎))
9291adantr 480 . . 3 ((𝜑𝐴 ∈ ω) → ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎))
93 pm4.61 404 . . . . 5 (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ↔ (𝐴𝑎 ∧ ¬ (𝐹‘suc 𝑎) = (𝐹𝑎)))
94 dfpss2 4111 . . . . . . 7 ((𝐹‘suc 𝑎) ⊊ (𝐹𝑎) ↔ ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) ∧ ¬ (𝐹‘suc 𝑎) = (𝐹𝑎)))
9594simplbi2 500 . . . . . 6 ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → (¬ (𝐹‘suc 𝑎) = (𝐹𝑎) → (𝐹‘suc 𝑎) ⊊ (𝐹𝑎)))
9695anim2d 611 . . . . 5 ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → ((𝐴𝑎 ∧ ¬ (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
9793, 96biimtrid 242 . . . 4 ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
9897ralimi 3089 . . 3 (∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → ∀𝑎 ∈ ω (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
99 rexim 3093 . . 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 206  wa 395  wo 846   = wceq 1537  wcel 2108  wral 3067  wrex 3076  Vcvv 3488  wss 3976  wpss 3977  𝒫 cpw 4622   cint 4970  ran crn 5701  Ord word 6394  suc csuc 6397   Fn wfn 6568  wf 6569  cfv 6573  ωcom 7903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-om 7904
This theorem is referenced by:  isf32lem5  10426
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