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Theorem isf32lem2 10349

Description: Lemma for isfin3-2 10362. 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 482	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ¬ ∩ ran 𝐹 ∈ ran 𝐹)
3		isf32lem.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺)
4	3	ffnd 6719	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn ω)
5		peano2 7881	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
6		fnfvelrn 7083	. . . . . . . . 9 ⊢ ((𝐹 Fn ω ∧ suc 𝐴 ∈ ω) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
7	4, 5, 6	syl2an 597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
8	7	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → (𝐹‘suc 𝐴) ∈ ran 𝐹)
9		intss1 4968	. . . . . . 7 ⊢ ((𝐹‘suc 𝐴) ∈ ran 𝐹 → ∩ ran 𝐹 ⊆ (𝐹‘suc 𝐴))
10	8, 9	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → ∩ ran 𝐹 ⊆ (𝐹‘suc 𝐴))
11		fvelrnb 6953	. . . . . . . . . . 11 ⊢ (𝐹 Fn ω → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹‘𝑐) = 𝑏))
12	4, 11	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹‘𝑐) = 𝑏))
13	12	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → (𝑏 ∈ ran 𝐹 ↔ ∃𝑐 ∈ ω (𝐹‘𝑐) = 𝑏))
14		simplrr 777	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴 ⊆ 𝑐) → 𝑐 ∈ ω)
15	5	ad3antlr 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴 ⊆ 𝑐) → suc 𝐴 ∈ ω)
16		simpr 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ suc 𝐴 ⊆ 𝑐) → suc 𝐴 ⊆ 𝑐)
17		simplrl 776	. . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = suc 𝐴 → (𝐹‘𝑏) = (𝐹‘suc 𝐴))
19	18	eqeq2d 2744	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = suc 𝐴 → ((𝐹‘suc 𝐴) = (𝐹‘𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘suc 𝐴)))
20	19	imbi2d 341	. . . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
22	21	eqeq2d 2744	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑑 → ((𝐹‘suc 𝐴) = (𝐹‘𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘𝑑)))
23	22	imbi2d 341	. . . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = suc 𝑑 → (𝐹‘𝑏) = (𝐹‘suc 𝑑))
25	24	eqeq2d 2744	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = suc 𝑑 → ((𝐹‘suc 𝐴) = (𝐹‘𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑)))
26	25	imbi2d 341	. . . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑐 → (𝐹‘𝑏) = (𝐹‘𝑐))
28	27	eqeq2d 2744	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑐 → ((𝐹‘suc 𝐴) = (𝐹‘𝑏) ↔ (𝐹‘suc 𝐴) = (𝐹‘𝑐)))
29	28	imbi2d 341	. . . . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . 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 3493	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (suc 𝐴 ∈ ω → suc 𝐴 ∈ V)
33		sucexb 7792	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
34	32, 33	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (suc 𝐴 ∈ ω → 𝐴 ∈ V)
35	34	adantl 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) → 𝐴 ∈ V)
36		sucssel 6460	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∈ V → (suc 𝐴 ⊆ 𝑑 → 𝐴 ∈ 𝑑))
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) → (suc 𝐴 ⊆ 𝑑 → 𝐴 ∈ 𝑑))
38	37	imp 408	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴 ⊆ 𝑑) → 𝐴 ∈ 𝑑)
39		eleq2w 2818	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑑 → (𝐴 ∈ 𝑎 ↔ 𝐴 ∈ 𝑑))
40		suceq 6431	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 𝑑 → suc 𝑎 = suc 𝑑)
41	40	fveq2d 6896	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑑 → (𝐹‘suc 𝑎) = (𝐹‘suc 𝑑))
42		fveq2 6892	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑑 → (𝐹‘𝑎) = (𝐹‘𝑑))
43	41, 42	eqeq12d 2749	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑑 → ((𝐹‘suc 𝑎) = (𝐹‘𝑎) ↔ (𝐹‘suc 𝑑) = (𝐹‘𝑑)))
44	39, 43	imbi12d 345	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑑 → ((𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ↔ (𝐴 ∈ 𝑑 → (𝐹‘suc 𝑑) = (𝐹‘𝑑))))
45	44	rspcv 3609	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ∈ ω → (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐴 ∈ 𝑑 → (𝐹‘suc 𝑑) = (𝐹‘𝑑))))
46	45	com23 86	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 ∈ ω → (𝐴 ∈ 𝑑 → (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐹‘suc 𝑑) = (𝐹‘𝑑))))
47	46	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 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 15	. . . . . . . . . . . . . . . . . . 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 2759	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹‘suc 𝐴) = (𝐹‘𝑑) ∧ (𝐹‘suc 𝑑) = (𝐹‘𝑑)) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑))
50	49	expcom 415	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘suc 𝑑) = (𝐹‘𝑑) → ((𝐹‘suc 𝐴) = (𝐹‘𝑑) → (𝐹‘suc 𝐴) = (𝐹‘suc 𝑑)))
51	48, 50	syl6 35	. . . . . . . . . . . . . . . . . 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 29	. . . . . . . . . . . . . . . . 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 7890	. . . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . 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 4041	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘suc 𝐴) = (𝐹‘𝑐) → (𝐹‘suc 𝐴) ⊆ (𝐹‘𝑐))
57	55, 56	syl 17	. . . . . . . . . . . . 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 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → suc 𝐴 ∈ ω)
59		simplrr 777	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝑐 ∈ ω)
60		simpr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) ∧ 𝑐 ⊆ suc 𝐴) → 𝑐 ⊆ suc 𝐴)
61		simplll 774	. . . . . . . . . . . . . 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 10348	. . . . . . . . . . . . . 14 ⊢ (((suc 𝐴 ∈ ω ∧ 𝑐 ∈ ω) ∧ (𝑐 ⊆ suc 𝐴 ∧ 𝜑)) → (𝐹‘suc 𝐴) ⊆ (𝐹‘𝑐))
64	58, 59, 60, 61, 63	syl22anc 838	. . . . . . . . . . . . 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 7863	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝐴 ∈ ω → Ord suc 𝐴)
66	5, 65	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ω → Ord suc 𝐴)
67	66	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) → Ord suc 𝐴)
68		nnord 7863	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ω → Ord 𝑐)
69	68	ad2antll 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) → Ord 𝑐)
70		ordtri2or2 6464	. . . . . . . . . . . . . 14 ⊢ ((Ord suc 𝐴 ∧ Ord 𝑐) → (suc 𝐴 ⊆ 𝑐 ∨ 𝑐 ⊆ suc 𝐴))
71	67, 69, 70	syl2anc 585	. . . . . . . . . . . . 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 958	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ∧ 𝑐 ∈ ω)) → (𝐹‘suc 𝐴) ⊆ (𝐹‘𝑐))
73	72	anassrs 469	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) ∧ 𝑐 ∈ ω) → (𝐹‘suc 𝐴) ⊆ (𝐹‘𝑐))
74		sseq2 4009	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑐) = 𝑏 → ((𝐹‘suc 𝐴) ⊆ (𝐹‘𝑐) ↔ (𝐹‘suc 𝐴) ⊆ 𝑏))
75	73, 74	syl5ibcom 244	. . . . . . . . . 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 3156	. . . . . . . . 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 239	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → (𝑏 ∈ ran 𝐹 → (𝐹‘suc 𝐴) ⊆ 𝑏))
78	77	ralrimiv 3146	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → ∀𝑏 ∈ ran 𝐹(𝐹‘suc 𝐴) ⊆ 𝑏)
79		ssint 4969	. . . . . . 7 ⊢ ((𝐹‘suc 𝐴) ⊆ ∩ ran 𝐹 ↔ ∀𝑏 ∈ ran 𝐹(𝐹‘suc 𝐴) ⊆ 𝑏)
80	78, 79	sylibr 233	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → (𝐹‘suc 𝐴) ⊆ ∩ ran 𝐹)
81	10, 80	eqssd 4000	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → ∩ ran 𝐹 = (𝐹‘suc 𝐴))
82	81, 8	eqeltrd 2834	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎))) → ∩ ran 𝐹 ∈ ran 𝐹)
83	2, 82	mtand 815	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ¬ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)))
84		rexnal 3101	. . 3 ⊢ (∃𝑎 ∈ ω ¬ (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ↔ ¬ ∀𝑎 ∈ ω (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)))
85	83, 84	sylibr 233	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∃𝑎 ∈ ω ¬ (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)))
86		suceq 6431	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
87	86	fveq2d 6896	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑎))
88		fveq2 6892	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
89	87, 88	sseq12d 4016	. . . . . 6 ⊢ (𝑥 = 𝑎 → ((𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎)))
90	89	cbvralvw 3235	. . . . 5 ⊢ (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) ↔ ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎))
91	62, 90	sylib 217	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎))
92	91	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∀𝑎 ∈ ω (𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎))
93		pm4.61 406	. . . . 5 ⊢ (¬ (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) ↔ (𝐴 ∈ 𝑎 ∧ ¬ (𝐹‘suc 𝑎) = (𝐹‘𝑎)))
94		dfpss2 4086	. . . . . . 7 ⊢ ((𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎) ↔ ((𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎) ∧ ¬ (𝐹‘suc 𝑎) = (𝐹‘𝑎)))
95	94	simplbi2 502	. . . . . 6 ⊢ ((𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎) → (¬ (𝐹‘suc 𝑎) = (𝐹‘𝑎) → (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎)))
96	95	anim2d 613	. . . . 5 ⊢ ((𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎) → ((𝐴 ∈ 𝑎 ∧ ¬ (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐴 ∈ 𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎))))
97	93, 96	biimtrid 241	. . . 4 ⊢ ((𝐹‘suc 𝑎) ⊆ (𝐹‘𝑎) → (¬ (𝐴 ∈ 𝑎 → (𝐹‘suc 𝑎) = (𝐹‘𝑎)) → (𝐴 ∈ 𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎))))
98	97	ralimi 3084	. . 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 3088	. . 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 18	. 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 15	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∃𝑎 ∈ ω (𝐴 ∈ 𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ⊆ wss 3949 ⊊ wpss 3950 𝒫 cpw 4603 ∩ cint 4951 ran crn 5678 Ord word 6364 suc csuc 6367 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 ωcom 7855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-om 7856

This theorem is referenced by: isf32lem5 10352

W3C validator