Theorem isf32lem2 10471

Description: Lemma for isfin3-2 10484. Non-minimum implies that there is always another decrease. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Hypotheses

Ref	Expression
isf32lem.a	⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺)
isf32lem.b	⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥))
isf32lem.c	⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹)

Assertion

Ref	Expression
isf32lem2	⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∃𝑎 ∈ ω (𝐴 ∈ 𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎)))

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.

Step	Hyp	Ref	Expression
1		isf32lem.c	. . . . 5 ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹)
2	1	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ¬ ∩ ran 𝐹 ∈ ran 𝐹)
3		isf32lem.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺)
4	3	ffnd 6740	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn ω)
5		peano2 7935	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
6		fnfvelrn 7112	. . . . . . . . 9 ⊢ ((𝐹 Fn ω ∧ suc 𝐴 ∈ ω) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
7	4, 5, 6	syl2an 608	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
8	7	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
9		intss1 4928	. . . . . . 7 ⊢ ((𝐹‘suc 𝐴) ∈ ran 𝐹 → ∩ ran 𝐹 ⊆ (𝐹‘suc 𝐴))
10	8, 9	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → ∩ ran 𝐹 ⊆ (𝐹‘suc 𝐴))
11		fvelrnb 6976	. . . . . . . . . . 11 ⊢ (𝐹 Fn ω → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹‘𝑐) = 𝑏))
12	4, 11	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹‘𝑐) = 𝑏))
13	12	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹‘𝑐) = 𝑏))
14		simplrr 790	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴 ⊆ 𝑐) → 𝑐 ∈ ω)
15	5	ad3antlr 744	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴 ⊆ 𝑐) → suc 𝐴 ∈ ω)
16		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴 ⊆ 𝑐) → suc 𝐴 ⊆ 𝑐)
17		simplrl 789	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴 ⊆ 𝑐) → ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)))
18		fveq2 6916	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = suc 𝐴 → (𝐹‘𝑏) = (𝐹‘suc 𝐴))
19	18	eqeq2d 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = suc 𝐴 → ((𝐹‘suc 𝐴) = (𝐹‘𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)))
20	19	imbi2d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = suc 𝐴 → ((∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘𝑏)) ↔ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴))))
21		fveq2 6916	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
22	21	eqeq2d 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑑 → ((𝐹‘suc 𝐴) = (𝐹‘𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘𝑑)))
23	22	imbi2d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑑 → ((∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘𝑏)) ↔ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘𝑑))))
24		fveq2 6916	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = suc 𝑑 → (𝐹‘𝑏) = (𝐹‘suc 𝑑))
25	24	eqeq2d 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = suc 𝑑 → ((𝐹‘suc 𝐴) = (𝐹‘𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑)))
26	25	imbi2d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = suc 𝑑 → ((∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘𝑏)) ↔ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))))
27		fveq2 6916	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑐 → (𝐹‘𝑏) = (𝐹‘𝑐))
28	27	eqeq2d 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑐 → ((𝐹‘suc 𝐴) = (𝐹‘𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘𝑐)))
29	28	imbi2d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑐 → ((∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘𝑏)) ↔ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘𝑐))))
30		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)
31	30	2a1i 12	. . . . . . . . . . . . . . . . 17 ⊢ (suc 𝐴 ∈ ω → (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)))
32		elex 3475	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (suc 𝐴 ∈ ω → suc 𝐴 ∈ V)
33		sucexb 7852	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
34	32, 33	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (suc 𝐴 ∈ ω → 𝐴 ∈ V)
35	34	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) → 𝐴 ∈ V)
36		sucssel 6486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∈ V → (suc 𝐴 ⊆ 𝑑 → 𝐴 ∈ 𝑑))
37	35, 36	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) → (suc 𝐴 ⊆ 𝑑 → 𝐴 ∈ 𝑑))
38	37	imp 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴 ⊆ 𝑑) → 𝐴 ∈ 𝑑)
39		eleq2w 2848	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑑 → (𝐴 ∈ 𝑎 ↔ 𝐴 ∈ 𝑑))
40		suceq 6457	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 𝑑 → suc 𝑎 = suc 𝑑)
41	40	fveq2d 6920	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑑 → (𝐹‘suc 𝑎) = (𝐹‘suc 𝑑))
42		fveq2 6916	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑑 → (𝐹‘𝑎) = (𝐹‘𝑑))
43	41, 42	eqeq12d 2779	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑑 → ((𝐹‘suc 𝑎) = (𝐹‘𝑎) ↔ (𝐹‘suc 𝑑) = (𝐹‘𝑑)))
44	39, 43	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑑 → ((𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ↔ (𝐴 ∈ 𝑑 → (𝐹‘suc 𝑑) = (𝐹‘𝑑))))
45	44	rspcv 3576	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ∈ ω → (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐴 ∈ 𝑑 → (𝐹‘suc 𝑑) = (𝐹‘𝑑))))
46	45	com23 87	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 ∈ ω → (𝐴 ∈ 𝑑 → (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝑑) = (𝐹‘𝑑))))
47	46	ad2antrr 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴 ⊆ 𝑑) → (𝐴 ∈ 𝑑 → (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝑑) = (𝐹‘𝑑))))
48	38, 47	mpd 16	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴 ⊆ 𝑑) → (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝑑) = (𝐹‘𝑑)))
49		eqtr3 2785	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹‘suc 𝐴) = (𝐹‘𝑑) ∧ (𝐹‘suc 𝑑) = (𝐹‘𝑑)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))
50	49	expcom 419	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘suc 𝑑) = (𝐹‘𝑑) → ((𝐹‘suc 𝐴) = (𝐹‘𝑑) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑)))
51	48, 50	syl6 36	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴 ⊆ 𝑑) → (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → ((𝐹‘suc 𝐴) = (𝐹‘𝑑) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))))
52	51	a2d 30	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴 ⊆ 𝑑) → ((∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘𝑑)) → (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))))
53	20, 23, 26, 29, 31, 52	findsg 7943	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴 ⊆ 𝑐) → (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝐴) = (𝐹‘𝑐)))
54	53	impr 460	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ (suc 𝐴 ⊆ 𝑐 ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)))) → (𝐹‘suc 𝐴) = (𝐹‘𝑐))
55	14, 15, 16, 17, 54	syl22anc 852	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴 ⊆ 𝑐) → (𝐹‘suc 𝐴) = (𝐹‘𝑐))
56		eqimss 3971	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘suc 𝐴) = (𝐹‘𝑐) → (𝐹‘suc 𝐴) ⊆ (𝐹‘𝑐))
57	55, 56	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴 ⊆ 𝑐) → (𝐹‘suc 𝐴) ⊆ (𝐹‘𝑐))
58	5	ad3antlr 744	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → suc 𝐴 ∈ ω)
59		simplrr 790	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝑐 ∈ ω)
60		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝑐 ⊆ suc 𝐴)
61		simplll 787	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝜑)
62		isf32lem.b	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥))
63	3, 62, 1	isf32lem1 10470	. . . . . . . . . . . . . 14 ⊢ (((suc 𝐴 ∈ ω ∧ 𝑐 ∈ ω) ∧ (𝑐 ⊆ suc 𝐴 ∧ 𝜑)) → (𝐹‘suc 𝐴) ⊆ (𝐹‘𝑐))
64	58, 59, 60, 61, 63	syl22anc 852	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → (𝐹‘suc 𝐴) ⊆ (𝐹‘𝑐))
65		nnord 7919	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝐴 ∈ ω → Ord suc 𝐴)
66	5, 65	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ω → Ord suc 𝐴)
67	66	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) → Ord suc 𝐴)
68		nnord 7919	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ω → Ord 𝑐)
69	68	ad2antll 742	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) → Ord 𝑐)
70		ordtri2or2 6490	. . . . . . . . . . . . . 14 ⊢ ((Ord suc 𝐴 ∧ Ord 𝑐) → (suc 𝐴 ⊆ 𝑐 ∨ 𝑐 ⊆ suc 𝐴))
71	67, 69, 70	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) → (suc 𝐴 ⊆ 𝑐 ∨ 𝑐 ⊆ suc 𝐴))
72	57, 64, 71	mpjaodan 973	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) → (𝐹‘suc 𝐴) ⊆ (𝐹‘𝑐))
73	72	anassrs 473	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) ∧ 𝑐 ∈ ω) → (𝐹‘suc 𝐴) ⊆ (𝐹‘𝑐))
74		sseq2 3939	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑐) = 𝑏 → ((𝐹‘suc 𝐴) ⊆ (𝐹‘𝑐) ↔ (𝐹‘suc 𝐴) ⊆ 𝑏))
75	73, 74	syl5ibcom 248	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) ∧ 𝑐 ∈ ω) → ((𝐹‘𝑐) = 𝑏 → (𝐹‘suc 𝐴) ⊆ 𝑏))
76	75	rexlimdva 3167	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → (∃𝑐 ∈ ω (𝐹‘𝑐) = 𝑏 → (𝐹‘suc 𝐴) ⊆ 𝑏))
77	13, 76	sylbid 243	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → (𝑏 ∈ ran 𝐹 → (𝐹‘suc 𝐴) ⊆ 𝑏))
78	77	ralrimiv 3157	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → ∀𝑏 ∈ ran 𝐹(𝐹‘suc 𝐴) ⊆ 𝑏)
79		ssint 4929	. . . . . . 7 ⊢ ((𝐹‘suc 𝐴) ⊆ ∩ ran 𝐹 ↔ ∀𝑏 ∈ ran 𝐹(𝐹‘suc 𝐴) ⊆ 𝑏)
80	78, 79	sylibr 237	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → (𝐹‘suc 𝐴) ⊆ ∩ ran 𝐹)
81	10, 80	eqssd 3930	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → ∩ ran 𝐹 = (𝐹‘suc 𝐴))
82	81, 8	eqeltrd 2864	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → ∩ ran 𝐹 ∈ ran 𝐹)
83	2, 82	mtand 828	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ¬ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)))
84		rexnal 3118	. . 3 ⊢ (∃𝑎 ∈ ω ¬ (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ↔ ¬ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)))
85	83, 84	sylibr 237	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∃𝑎 ∈ ω ¬ (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)))
86		suceq 6457	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
87	86	fveq2d 6920	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑎))
88		fveq2 6916	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
89	87, 88	sseq12d 3946	. . . . . 6 ⊢ (𝑥 = 𝑎 → ((𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎)))
90	89	cbvralvw 3244	. . . . 5 ⊢ (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) ↔ ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎))
91	62, 90	sylib 221	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎))
92	91	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎))
93		pm4.61 410	. . . . 5 ⊢ (¬ (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ↔ (𝐴 ∈ 𝑎 ∧ ¬ (𝐹‘suc 𝑎) = (𝐹‘𝑎)))
94		dfpss2 4018	. . . . . . 7 ⊢ ((𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎) ↔ ((𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎) ∧ ¬ (𝐹‘suc 𝑎) = (𝐹‘𝑎)))
95	94	simplbi2 506	. . . . . 6 ⊢ ((𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎) → (¬ (𝐹‘suc 𝑎) = (𝐹‘𝑎) → (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎)))
96	95	anim2d 624	. . . . 5 ⊢ ((𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎) → ((𝐴 ∈ 𝑎 ∧ ¬ (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐴 ∈ 𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎))))
97	93, 96	biimtrid 245	. . . 4 ⊢ ((𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎) → (¬ (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐴 ∈ 𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎))))
98	97	ralimi 3103	. . 3 ⊢ (∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎) → ∀𝑎 ∈ ω (¬ (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐴 ∈ 𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎))))
99		rexim 3107	. . 3 ⊢ (∀𝑎 ∈ ω (¬ (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐴 ∈ 𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎))) → (∃𝑎 ∈ ω ¬ (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → ∃𝑎 ∈ ω (𝐴 ∈ 𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎))))
100	92, 98, 99	3syl 19	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (∃𝑎 ∈ ω ¬ (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → ∃𝑎 ∈ ω (𝐴 ∈ 𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎))))
101	85, 100	mpd 16	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∃𝑎 ∈ ω (𝐴 ∈ 𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ⊊ wpss 4016  𝒫 cpw 4560  ∩ cint 4912  ran crn 5659  Ord word 6387  suc csuc 6390   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  ωcom 7911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-om 7912

This theorem is used by:  isf32lem5  10474