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Theorem isf32lem2 9120
Description: Lemma for isfin3-2 9133. 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 481 . . . 4 ((𝜑𝐴 ∈ ω) → ¬ ran 𝐹 ∈ ran 𝐹)
3 isf32lem.a . . . . . . . . . 10 (𝜑𝐹:ω⟶𝒫 𝐺)
4 ffn 6002 . . . . . . . . . 10 (𝐹:ω⟶𝒫 𝐺𝐹 Fn ω)
53, 4syl 17 . . . . . . . . 9 (𝜑𝐹 Fn ω)
6 peano2 7033 . . . . . . . . 9 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
7 fnfvelrn 6312 . . . . . . . . 9 ((𝐹 Fn ω ∧ suc 𝐴 ∈ ω) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
85, 6, 7syl2an 494 . . . . . . . 8 ((𝜑𝐴 ∈ ω) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
98adantr 481 . . . . . . 7 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
10 intss1 4457 . . . . . . 7 ((𝐹‘suc 𝐴) ∈ ran 𝐹 ran 𝐹 ⊆ (𝐹‘suc 𝐴))
119, 10syl 17 . . . . . 6 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ran 𝐹 ⊆ (𝐹‘suc 𝐴))
12 fvelrnb 6200 . . . . . . . . . . 11 (𝐹 Fn ω → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹𝑐) = 𝑏))
135, 12syl 17 . . . . . . . . . 10 (𝜑 → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹𝑐) = 𝑏))
1413ad2antrr 761 . . . . . . . . 9 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹𝑐) = 𝑏))
15 simplrr 800 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → 𝑐 ∈ ω)
166ad3antlr 766 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → suc 𝐴 ∈ ω)
17 simpr 477 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → suc 𝐴𝑐)
18 simplrl 799 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
19 fveq2 6148 . . . . . . . . . . . . . . . . . . 19 (𝑏 = suc 𝐴 → (𝐹𝑏) = (𝐹‘suc 𝐴))
2019eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑏 = suc 𝐴 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)))
2120imbi2d 330 . . . . . . . . . . . . . . . . 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))))
22 fveq2 6148 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑑 → (𝐹𝑏) = (𝐹𝑑))
2322eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑑 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹𝑑)))
2423imbi2d 330 . . . . . . . . . . . . . . . . 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))))
25 fveq2 6148 . . . . . . . . . . . . . . . . . . 19 (𝑏 = suc 𝑑 → (𝐹𝑏) = (𝐹‘suc 𝑑))
2625eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑏 = suc 𝑑 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑)))
2726imbi2d 330 . . . . . . . . . . . . . . . . 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))))
28 fveq2 6148 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑐 → (𝐹𝑏) = (𝐹𝑐))
2928eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑐 → ((𝐹‘suc 𝐴) = (𝐹𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹𝑐)))
3029imbi2d 330 . . . . . . . . . . . . . . . . 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))))
31 eqid 2621 . . . . . . . . . . . . . . . . . 18 (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)
32312a1i 12 . . . . . . . . . . . . . . . . 17 (suc 𝐴 ∈ ω → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)))
33 elex 3198 . . . . . . . . . . . . . . . . . . . . . . . 24 (suc 𝐴 ∈ ω → suc 𝐴 ∈ V)
34 sucexb 6956 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3533, 34sylibr 224 . . . . . . . . . . . . . . . . . . . . . . 23 (suc 𝐴 ∈ ω → 𝐴 ∈ V)
3635adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) → 𝐴 ∈ V)
37 sucssel 5778 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ V → (suc 𝐴𝑑𝐴𝑑))
3836, 37syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) → (suc 𝐴𝑑𝐴𝑑))
3938imp 445 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → 𝐴𝑑)
40 eleq2 2687 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑑 → (𝐴𝑎𝐴𝑑))
41 suceq 5749 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑑 → suc 𝑎 = suc 𝑑)
4241fveq2d 6152 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑑 → (𝐹‘suc 𝑎) = (𝐹‘suc 𝑑))
43 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑑 → (𝐹𝑎) = (𝐹𝑑))
4442, 43eqeq12d 2636 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑑 → ((𝐹‘suc 𝑎) = (𝐹𝑎) ↔ (𝐹‘suc 𝑑) = (𝐹𝑑)))
4540, 44imbi12d 334 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝑑 → ((𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ↔ (𝐴𝑑 → (𝐹‘suc 𝑑) = (𝐹𝑑))))
4645rspcv 3291 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ ω → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑑 → (𝐹‘suc 𝑑) = (𝐹𝑑))))
4746com23 86 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 ∈ ω → (𝐴𝑑 → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝑑) = (𝐹𝑑))))
4847ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → (𝐴𝑑 → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝑑) = (𝐹𝑑))))
4939, 48mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑑) → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝑑) = (𝐹𝑑)))
50 eqtr3 2642 . . . . . . . . . . . . . . . . . . . 20 (((𝐹‘suc 𝐴) = (𝐹𝑑) ∧ (𝐹‘suc 𝑑) = (𝐹𝑑)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))
5150expcom 451 . . . . . . . . . . . . . . . . . . 19 ((𝐹‘suc 𝑑) = (𝐹𝑑) → ((𝐹‘suc 𝐴) = (𝐹𝑑) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑)))
5249, 51syl6 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))))
5352a2d 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))))
5421, 24, 27, 30, 32, 53findsg 7040 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑐) → (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐹‘suc 𝐴) = (𝐹𝑐)))
5554impr 648 . . . . . . . . . . . . . . 15 (((𝑐 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ (suc 𝐴𝑐 ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))) → (𝐹‘suc 𝐴) = (𝐹𝑐))
5615, 16, 17, 18, 55syl22anc 1324 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → (𝐹‘suc 𝐴) = (𝐹𝑐))
57 eqimss 3636 . . . . . . . . . . . . . 14 ((𝐹‘suc 𝐴) = (𝐹𝑐) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
5856, 57syl 17 . . . . . . . . . . . . 13 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴𝑐) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
596ad3antlr 766 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → suc 𝐴 ∈ ω)
60 simplrr 800 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝑐 ∈ ω)
61 simpr 477 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝑐 ⊆ suc 𝐴)
62 simplll 797 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝜑)
63 isf32lem.b . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))
643, 63, 1isf32lem1 9119 . . . . . . . . . . . . . 14 (((suc 𝐴 ∈ ω ∧ 𝑐 ∈ ω) ∧ (𝑐 ⊆ suc 𝐴𝜑)) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
6559, 60, 61, 62, 64syl22anc 1324 . . . . . . . . . . . . 13 ((((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
66 nnord 7020 . . . . . . . . . . . . . . . 16 (suc 𝐴 ∈ ω → Ord suc 𝐴)
676, 66syl 17 . . . . . . . . . . . . . . 15 (𝐴 ∈ ω → Ord suc 𝐴)
6867ad2antlr 762 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → Ord suc 𝐴)
69 nnord 7020 . . . . . . . . . . . . . . 15 (𝑐 ∈ ω → Ord 𝑐)
7069ad2antll 764 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → Ord 𝑐)
71 ordtri2or2 5782 . . . . . . . . . . . . . 14 ((Ord suc 𝐴 ∧ Ord 𝑐) → (suc 𝐴𝑐𝑐 ⊆ suc 𝐴))
7268, 70, 71syl2anc 692 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → (suc 𝐴𝑐𝑐 ⊆ suc 𝐴))
7358, 65, 72mpjaodan 826 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ∧ 𝑐 ∈ ω)) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
7473anassrs 679 . . . . . . . . . . 11 ((((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) ∧ 𝑐 ∈ ω) → (𝐹‘suc 𝐴) ⊆ (𝐹𝑐))
75 sseq2 3606 . . . . . . . . . . 11 ((𝐹𝑐) = 𝑏 → ((𝐹‘suc 𝐴) ⊆ (𝐹𝑐) ↔ (𝐹‘suc 𝐴) ⊆ 𝑏))
7674, 75syl5ibcom 235 . . . . . . . . . 10 ((((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) ∧ 𝑐 ∈ ω) → ((𝐹𝑐) = 𝑏 → (𝐹‘suc 𝐴) ⊆ 𝑏))
7776rexlimdva 3024 . . . . . . . . 9 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (∃𝑐 ∈ ω (𝐹𝑐) = 𝑏 → (𝐹‘suc 𝐴) ⊆ 𝑏))
7814, 77sylbid 230 . . . . . . . 8 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝑏 ∈ ran 𝐹 → (𝐹‘suc 𝐴) ⊆ 𝑏))
7978ralrimiv 2959 . . . . . . 7 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ∀𝑏 ∈ ran 𝐹(𝐹‘suc 𝐴) ⊆ 𝑏)
80 ssint 4458 . . . . . . 7 ((𝐹‘suc 𝐴) ⊆ ran 𝐹 ↔ ∀𝑏 ∈ ran 𝐹(𝐹‘suc 𝐴) ⊆ 𝑏)
8179, 80sylibr 224 . . . . . 6 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → (𝐹‘suc 𝐴) ⊆ ran 𝐹)
8211, 81eqssd 3600 . . . . 5 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ran 𝐹 = (𝐹‘suc 𝐴))
8382, 9eqeltrd 2698 . . . 4 (((𝜑𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎))) → ran 𝐹 ∈ ran 𝐹)
842, 83mtand 690 . . 3 ((𝜑𝐴 ∈ ω) → ¬ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
85 rexnal 2989 . . 3 (∃𝑎 ∈ ω ¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ↔ ¬ ∀𝑎 ∈ ω (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
8684, 85sylibr 224 . 2 ((𝜑𝐴 ∈ ω) → ∃𝑎 ∈ ω ¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)))
87 suceq 5749 . . . . . . . 8 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
8887fveq2d 6152 . . . . . . 7 (𝑥 = 𝑎 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑎))
89 fveq2 6148 . . . . . . 7 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
9088, 89sseq12d 3613 . . . . . 6 (𝑥 = 𝑎 → ((𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ↔ (𝐹‘suc 𝑎) ⊆ (𝐹𝑎)))
9190cbvralv 3159 . . . . 5 (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ↔ ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎))
9263, 91sylib 208 . . . 4 (𝜑 → ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎))
9392adantr 481 . . 3 ((𝜑𝐴 ∈ ω) → ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎))
94 pm4.61 442 . . . . 5 (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) ↔ (𝐴𝑎 ∧ ¬ (𝐹‘suc 𝑎) = (𝐹𝑎)))
95 dfpss2 3670 . . . . . . 7 ((𝐹‘suc 𝑎) ⊊ (𝐹𝑎) ↔ ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) ∧ ¬ (𝐹‘suc 𝑎) = (𝐹𝑎)))
9695simplbi2 654 . . . . . 6 ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → (¬ (𝐹‘suc 𝑎) = (𝐹𝑎) → (𝐹‘suc 𝑎) ⊊ (𝐹𝑎)))
9796anim2d 588 . . . . 5 ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → ((𝐴𝑎 ∧ ¬ (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
9894, 97syl5bi 232 . . . 4 ((𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
9998ralimi 2947 . . 3 (∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹𝑎) → ∀𝑎 ∈ ω (¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
100 rexim 3002 . . 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))))
10193, 99, 1003syl 18 . 2 ((𝜑𝐴 ∈ ω) → (∃𝑎 ∈ ω ¬ (𝐴𝑎 → (𝐹‘suc 𝑎) = (𝐹𝑎)) → ∃𝑎 ∈ ω (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎))))
10286, 101mpd 15 1 ((𝜑𝐴 ∈ ω) → ∃𝑎 ∈ ω (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  wss 3555  wpss 3556  𝒫 cpw 4130   cint 4440  ran crn 5075  Ord word 5681  suc csuc 5684   Fn wfn 5842  wf 5843  cfv 5847  ωcom 7012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-om 7013
This theorem is referenced by:  isf32lem5  9123
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