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Theorem infpssrlem4 10285

Description: Lemma for infpssr 10287. (Contributed by Stefan O'Rear, 30-Oct-2014.)

**Hypotheses**
Ref	Expression
infpssrlem.a	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
infpssrlem.c	⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐴)
infpssrlem.d	⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵))
infpssrlem.e	⊢ 𝐺 = (rec(^◡𝐹, 𝐶) ↾ ω)

**Assertion**
Ref	Expression
infpssrlem4	⊢ ((𝜑 ∧ 𝑀 ∈ ω ∧ 𝑁 ∈ 𝑀) → (𝐺‘𝑀) ≠ (𝐺‘𝑁))

Proof of Theorem infpssrlem4 Dummy variables 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . . . . . 8 ⊢ (𝑐 = ∅ → (𝐺‘𝑐) = (𝐺‘∅))
2	1	neeq1d 3017	. . . . . . 7 ⊢ (𝑐 = ∅ → ((𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ (𝐺‘∅) ≠ (𝐺‘𝑏)))
3	2	raleqbi1dv 3333	. . . . . 6 ⊢ (𝑐 = ∅ → (∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ ∀𝑏 ∈ ∅ (𝐺‘∅) ≠ (𝐺‘𝑏)))
4	3	imbi2d 343	. . . . 5 ⊢ (𝑐 = ∅ → ((𝜑 → ∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏)) ↔ (𝜑 → ∀𝑏 ∈ ∅ (𝐺‘∅) ≠ (𝐺‘𝑏))))
5		fveq2 6881	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (𝐺‘𝑐) = (𝐺‘𝑑))
6	5	neeq1d 3017	. . . . . . 7 ⊢ (𝑐 = 𝑑 → ((𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ (𝐺‘𝑑) ≠ (𝐺‘𝑏)))
7	6	raleqbi1dv 3333	. . . . . 6 ⊢ (𝑐 = 𝑑 → (∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)))
8	7	imbi2d 343	. . . . 5 ⊢ (𝑐 = 𝑑 → ((𝜑 → ∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏)) ↔ (𝜑 → ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏))))
9		fveq2 6881	. . . . . . . 8 ⊢ (𝑐 = suc 𝑑 → (𝐺‘𝑐) = (𝐺‘suc 𝑑))
10	9	neeq1d 3017	. . . . . . 7 ⊢ (𝑐 = suc 𝑑 → ((𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ (𝐺‘suc 𝑑) ≠ (𝐺‘𝑏)))
11	10	raleqbi1dv 3333	. . . . . 6 ⊢ (𝑐 = suc 𝑑 → (∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ ∀𝑏 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑏)))
12	11	imbi2d 343	. . . . 5 ⊢ (𝑐 = suc 𝑑 → ((𝜑 → ∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏)) ↔ (𝜑 → ∀𝑏 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑏))))
13		fveq2 6881	. . . . . . . 8 ⊢ (𝑐 = 𝑀 → (𝐺‘𝑐) = (𝐺‘𝑀))
14	13	neeq1d 3017	. . . . . . 7 ⊢ (𝑐 = 𝑀 → ((𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ (𝐺‘𝑀) ≠ (𝐺‘𝑏)))
15	14	raleqbi1dv 3333	. . . . . 6 ⊢ (𝑐 = 𝑀 → (∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ ∀𝑏 ∈ 𝑀 (𝐺‘𝑀) ≠ (𝐺‘𝑏)))
16	15	imbi2d 343	. . . . 5 ⊢ (𝑐 = 𝑀 → ((𝜑 → ∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏)) ↔ (𝜑 → ∀𝑏 ∈ 𝑀 (𝐺‘𝑀) ≠ (𝐺‘𝑏))))
17		ral0 4459	. . . . . 6 ⊢ ∀𝑏 ∈ ∅ (𝐺‘∅) ≠ (𝐺‘𝑏)
18	17	a1i 11	. . . . 5 ⊢ (𝜑 → ∀𝑏 ∈ ∅ (𝐺‘∅) ≠ (𝐺‘𝑏))
19		infpssrlem.c	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐴)
20		f1ocnv 6833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝐵–1-1-onto→𝐴 → ^◡𝐹:𝐴–1-1-onto→𝐵)
21		f1of 6820	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡𝐹:𝐴–1-1-onto→𝐵 → ^◡𝐹:𝐴⟶𝐵)
22	19, 20, 21	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ^◡𝐹:𝐴⟶𝐵)
23	22	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → ^◡𝐹:𝐴⟶𝐵)
24		infpssrlem.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
25		infpssrlem.d	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵))
26		infpssrlem.e	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐺 = (rec(^◡𝐹, 𝐶) ↾ ω)
27	24, 19, 25, 26	infpssrlem3 10284	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺:ω⟶𝐴)
28	27	ffvelcdmda 7079	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → (𝐺‘𝑑) ∈ 𝐴)
29	28	ancoms 463	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → (𝐺‘𝑑) ∈ 𝐴)
30	23, 29	ffvelcdmd 7080	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → (^◡𝐹‘(𝐺‘𝑑)) ∈ 𝐵)
31	25	eldifbd 3918	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)
32	31	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → ¬ 𝐶 ∈ 𝐵)
33		nelne2 3056	. . . . . . . . . . . . . . 15 ⊢ (((^◡𝐹‘(𝐺‘𝑑)) ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → (^◡𝐹‘(𝐺‘𝑑)) ≠ 𝐶)
34	30, 32, 33	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → (^◡𝐹‘(𝐺‘𝑑)) ≠ 𝐶)
35	24, 19, 25, 26	infpssrlem2 10283	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ ω → (𝐺‘suc 𝑑) = (^◡𝐹‘(𝐺‘𝑑)))
36	35	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → (𝐺‘suc 𝑑) = (^◡𝐹‘(𝐺‘𝑑)))
37	24, 19, 25, 26	infpssrlem1 10282	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
38	37	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → (𝐺‘∅) = 𝐶)
39	34, 36, 38	3netr4d 3035	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → (𝐺‘suc 𝑑) ≠ (𝐺‘∅))
40	39	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) → (𝐺‘suc 𝑑) ≠ (𝐺‘∅))
41	1	neeq2d 3018	. . . . . . . . . . . 12 ⊢ (𝑐 = ∅ → ((𝐺‘suc 𝑑) ≠ (𝐺‘𝑐) ↔ (𝐺‘suc 𝑑) ≠ (𝐺‘∅)))
42	40, 41	imbitrrid 249	. . . . . . . . . . 11 ⊢ (𝑐 = ∅ → ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐)))
43	42	adantrd 496	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑) → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐)))
44		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∈ ω ∧ 𝑐 ∈ suc 𝑑) → 𝑐 ∈ suc 𝑑)
45		peano2 7882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ ω → suc 𝑑 ∈ ω)
46	45	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∈ ω ∧ 𝑐 ∈ suc 𝑑) → suc 𝑑 ∈ ω)
47		elnn 7869	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ suc 𝑑 ∧ suc 𝑑 ∈ ω) → 𝑐 ∈ ω)
48	44, 46, 47	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ ω ∧ 𝑐 ∈ suc 𝑑) → 𝑐 ∈ ω)
49	48	3ad2antl1 1204	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑) → 𝑐 ∈ ω)
50	49	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝑐 ≠ ∅ ∧ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑)) → 𝑐 ∈ ω)
51		simpl 487	. . . . . . . . . . . . 13 ⊢ ((𝑐 ≠ ∅ ∧ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑)) → 𝑐 ≠ ∅)
52		nnsuc 7876	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ ω ∧ 𝑐 ≠ ∅) → ∃𝑏 ∈ ω 𝑐 = suc 𝑏)
53	50, 51, 52	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝑐 ≠ ∅ ∧ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑)) → ∃𝑏 ∈ ω 𝑐 = suc 𝑏)
54		nfv 1944	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑏 𝑑 ∈ ω
55		nfv 1944	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑏𝜑
56		nfra1 3289	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑏∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)
57	54, 55, 56	nf3an 1931	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑏(𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏))
58		nfv 1944	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑏 𝑐 ∈ suc 𝑑
59	57, 58	nfan 1929	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑)
60		nfv 1944	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏(𝐺‘suc 𝑑) ≠ (𝐺‘𝑐)
61		simpl3 1212	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ (suc 𝑏 ∈ suc 𝑑 ∧ 𝑏 ∈ ω)) → ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏))
62		simpr 489	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑑 ∈ ω ∧ suc 𝑏 ∈ suc 𝑑) → suc 𝑏 ∈ suc 𝑑)
63		nnord 7866	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑑 ∈ ω → Ord 𝑑)
64	63	adantr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑑 ∈ ω ∧ suc 𝑏 ∈ suc 𝑑) → Ord 𝑑)
65		ordsucelsuc 7814	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Ord 𝑑 → (𝑏 ∈ 𝑑 ↔ suc 𝑏 ∈ suc 𝑑))
66	64, 65	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑑 ∈ ω ∧ suc 𝑏 ∈ suc 𝑑) → (𝑏 ∈ 𝑑 ↔ suc 𝑏 ∈ suc 𝑑))
67	62, 66	mpbird 260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑑 ∈ ω ∧ suc 𝑏 ∈ suc 𝑑) → 𝑏 ∈ 𝑑)
68	67	3ad2antl1 1204	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ suc 𝑏 ∈ suc 𝑑) → 𝑏 ∈ 𝑑)
69	68	adantrr 729	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ (suc 𝑏 ∈ suc 𝑑 ∧ 𝑏 ∈ ω)) → 𝑏 ∈ 𝑑)
70		rsp 3253	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏) → (𝑏 ∈ 𝑑 → (𝐺‘𝑑) ≠ (𝐺‘𝑏)))
71	61, 69, 70	sylc 66	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ (suc 𝑏 ∈ suc 𝑑 ∧ 𝑏 ∈ ω)) → (𝐺‘𝑑) ≠ (𝐺‘𝑏))
72		f1of1 6819	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (^◡𝐹:𝐴–1-1-onto→𝐵 → ^◡𝐹:𝐴–1-1→𝐵)
73	19, 20, 72	3syl 19	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ^◡𝐹:𝐴–1-1→𝐵)
74	73	ad2antlr 739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → ^◡𝐹:𝐴–1-1→𝐵)
75	29	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → (𝐺‘𝑑) ∈ 𝐴)
76	27	ffvelcdmda 7079	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑏 ∈ ω) → (𝐺‘𝑏) ∈ 𝐴)
77	76	adantll 726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → (𝐺‘𝑏) ∈ 𝐴)
78		f1fveq 7260	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((^◡𝐹:𝐴–1-1→𝐵 ∧ ((𝐺‘𝑑) ∈ 𝐴 ∧ (𝐺‘𝑏) ∈ 𝐴)) → ((^◡𝐹‘(𝐺‘𝑑)) = (^◡𝐹‘(𝐺‘𝑏)) ↔ (𝐺‘𝑑) = (𝐺‘𝑏)))
79	74, 75, 77, 78	syl12anc 849	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → ((^◡𝐹‘(𝐺‘𝑑)) = (^◡𝐹‘(𝐺‘𝑏)) ↔ (𝐺‘𝑑) = (𝐺‘𝑏)))
80	79	necon3bid 3002	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → ((^◡𝐹‘(𝐺‘𝑑)) ≠ (^◡𝐹‘(𝐺‘𝑏)) ↔ (𝐺‘𝑑) ≠ (𝐺‘𝑏)))
81	80	biimprd 251	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → ((𝐺‘𝑑) ≠ (𝐺‘𝑏) → (^◡𝐹‘(𝐺‘𝑑)) ≠ (^◡𝐹‘(𝐺‘𝑏))))
82	35	adantr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑑 ∈ ω ∧ 𝑏 ∈ ω) → (𝐺‘suc 𝑑) = (^◡𝐹‘(𝐺‘𝑑)))
83	24, 19, 25, 26	infpssrlem2 10283	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 ∈ ω → (𝐺‘suc 𝑏) = (^◡𝐹‘(𝐺‘𝑏)))
84	83	adantl 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑑 ∈ ω ∧ 𝑏 ∈ ω) → (𝐺‘suc 𝑏) = (^◡𝐹‘(𝐺‘𝑏)))
85	82, 84	neeq12d 3019	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑑 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏) ↔ (^◡𝐹‘(𝐺‘𝑑)) ≠ (^◡𝐹‘(𝐺‘𝑏))))
86	85	adantlr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → ((𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏) ↔ (^◡𝐹‘(𝐺‘𝑑)) ≠ (^◡𝐹‘(𝐺‘𝑏))))
87	81, 86	sylibrd 262	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → ((𝐺‘𝑑) ≠ (𝐺‘𝑏) → (𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏)))
88	87	adantrl 728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ (suc 𝑏 ∈ suc 𝑑 ∧ 𝑏 ∈ ω)) → ((𝐺‘𝑑) ≠ (𝐺‘𝑏) → (𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏)))
89	88	3adantl3 1187	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ (suc 𝑏 ∈ suc 𝑑 ∧ 𝑏 ∈ ω)) → ((𝐺‘𝑑) ≠ (𝐺‘𝑏) → (𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏)))
90	71, 89	mpd 16	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ (suc 𝑏 ∈ suc 𝑑 ∧ 𝑏 ∈ ω)) → (𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏))
91	90	expr 461	. . . . . . . . . . . . . . . 16 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ suc 𝑏 ∈ suc 𝑑) → (𝑏 ∈ ω → (𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏)))
92		eleq1 2851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = suc 𝑏 → (𝑐 ∈ suc 𝑑 ↔ suc 𝑏 ∈ suc 𝑑))
93	92	anbi2d 641	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = suc 𝑏 → (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑) ↔ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ suc 𝑏 ∈ suc 𝑑)))
94		fveq2 6881	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = suc 𝑏 → (𝐺‘𝑐) = (𝐺‘suc 𝑏))
95	94	neeq2d 3018	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = suc 𝑏 → ((𝐺‘suc 𝑑) ≠ (𝐺‘𝑐) ↔ (𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏)))
96	95	imbi2d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = suc 𝑏 → ((𝑏 ∈ ω → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐)) ↔ (𝑏 ∈ ω → (𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏))))
97	93, 96	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = suc 𝑏 → ((((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑) → (𝑏 ∈ ω → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐))) ↔ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ suc 𝑏 ∈ suc 𝑑) → (𝑏 ∈ ω → (𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏)))))
98	91, 97	mpbiri 261	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = suc 𝑏 → (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑) → (𝑏 ∈ ω → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐))))
99	98	com3l 90	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑) → (𝑏 ∈ ω → (𝑐 = suc 𝑏 → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐))))
100	59, 60, 99	rexlimd 3272	. . . . . . . . . . . . 13 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑) → (∃𝑏 ∈ ω 𝑐 = suc 𝑏 → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐)))
101	100	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑐 ≠ ∅ ∧ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑)) → (∃𝑏 ∈ ω 𝑐 = suc 𝑏 → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐)))
102	53, 101	mpd 16	. . . . . . . . . . 11 ⊢ ((𝑐 ≠ ∅ ∧ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑)) → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐))
103	102	ex 417	. . . . . . . . . 10 ⊢ (𝑐 ≠ ∅ → (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑) → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐)))
104	43, 103	pm2.61ine 3041	. . . . . . . . 9 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑) → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐))
105	104	ralrimiva 3157	. . . . . . . 8 ⊢ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) → ∀𝑐 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑐))
106		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → (𝐺‘𝑐) = (𝐺‘𝑏))
107	106	neeq2d 3018	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → ((𝐺‘suc 𝑑) ≠ (𝐺‘𝑐) ↔ (𝐺‘suc 𝑑) ≠ (𝐺‘𝑏)))
108	107	cbvralvw 3243	. . . . . . . 8 ⊢ (∀𝑐 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑐) ↔ ∀𝑏 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑏))
109	105, 108	sylib 221	. . . . . . 7 ⊢ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) → ∀𝑏 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑏))
110	109	3exp 1137	. . . . . 6 ⊢ (𝑑 ∈ ω → (𝜑 → (∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏) → ∀𝑏 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑏))))
111	110	a2d 30	. . . . 5 ⊢ (𝑑 ∈ ω → ((𝜑 → ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) → (𝜑 → ∀𝑏 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑏))))
112	4, 8, 12, 16, 18, 111	finds 7889	. . . 4 ⊢ (𝑀 ∈ ω → (𝜑 → ∀𝑏 ∈ 𝑀 (𝐺‘𝑀) ≠ (𝐺‘𝑏)))
113	112	impcom 412	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ ω) → ∀𝑏 ∈ 𝑀 (𝐺‘𝑀) ≠ (𝐺‘𝑏))
114		fveq2 6881	. . . . 5 ⊢ (𝑏 = 𝑁 → (𝐺‘𝑏) = (𝐺‘𝑁))
115	114	neeq2d 3018	. . . 4 ⊢ (𝑏 = 𝑁 → ((𝐺‘𝑀) ≠ (𝐺‘𝑏) ↔ (𝐺‘𝑀) ≠ (𝐺‘𝑁)))
116	115	rspccv 3578	. . 3 ⊢ (∀𝑏 ∈ 𝑀 (𝐺‘𝑀) ≠ (𝐺‘𝑏) → (𝑁 ∈ 𝑀 → (𝐺‘𝑀) ≠ (𝐺‘𝑁)))
117	113, 116	syl 18	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ ω) → (𝑁 ∈ 𝑀 → (𝐺‘𝑀) ≠ (𝐺‘𝑁)))
118	117	3impia 1135	1 ⊢ ((𝜑 ∧ 𝑀 ∈ ω ∧ 𝑁 ∈ 𝑀) → (𝐺‘𝑀) ≠ (𝐺‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 ∖ cdif 3902 ⊆ wss 3905 ∅c0 4286 ^◡ccnv 5660 ↾ cres 5663 Ord word 6359 suc csuc 6362 ⟶wf 6532 –1-1→wf1 6533 –1-1-onto→wf1o 6535 ‘cfv 6536 ωcom 7858 reccrdg 8392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-om 7859 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393

This theorem is referenced by: infpssrlem5 10286

W3C validator