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Theorem infpssrlem4 9993

Description: Lemma for infpssr 9995. (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 6756	. . . . . . . 8 ⊢ (𝑐 = ∅ → (𝐺‘𝑐) = (𝐺‘∅))
2	1	neeq1d 3002	. . . . . . 7 ⊢ (𝑐 = ∅ → ((𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ (𝐺‘∅) ≠ (𝐺‘𝑏)))
3	2	raleqbi1dv 3331	. . . . . 6 ⊢ (𝑐 = ∅ → (∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ ∀𝑏 ∈ ∅ (𝐺‘∅) ≠ (𝐺‘𝑏)))
4	3	imbi2d 340	. . . . 5 ⊢ (𝑐 = ∅ → ((𝜑 → ∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏)) ↔ (𝜑 → ∀𝑏 ∈ ∅ (𝐺‘∅) ≠ (𝐺‘𝑏))))
5		fveq2 6756	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (𝐺‘𝑐) = (𝐺‘𝑑))
6	5	neeq1d 3002	. . . . . . 7 ⊢ (𝑐 = 𝑑 → ((𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ (𝐺‘𝑑) ≠ (𝐺‘𝑏)))
7	6	raleqbi1dv 3331	. . . . . 6 ⊢ (𝑐 = 𝑑 → (∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)))
8	7	imbi2d 340	. . . . 5 ⊢ (𝑐 = 𝑑 → ((𝜑 → ∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏)) ↔ (𝜑 → ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏))))
9		fveq2 6756	. . . . . . . 8 ⊢ (𝑐 = suc 𝑑 → (𝐺‘𝑐) = (𝐺‘suc 𝑑))
10	9	neeq1d 3002	. . . . . . 7 ⊢ (𝑐 = suc 𝑑 → ((𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ (𝐺‘suc 𝑑) ≠ (𝐺‘𝑏)))
11	10	raleqbi1dv 3331	. . . . . 6 ⊢ (𝑐 = suc 𝑑 → (∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ ∀𝑏 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑏)))
12	11	imbi2d 340	. . . . 5 ⊢ (𝑐 = suc 𝑑 → ((𝜑 → ∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏)) ↔ (𝜑 → ∀𝑏 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑏))))
13		fveq2 6756	. . . . . . . 8 ⊢ (𝑐 = 𝑀 → (𝐺‘𝑐) = (𝐺‘𝑀))
14	13	neeq1d 3002	. . . . . . 7 ⊢ (𝑐 = 𝑀 → ((𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ (𝐺‘𝑀) ≠ (𝐺‘𝑏)))
15	14	raleqbi1dv 3331	. . . . . 6 ⊢ (𝑐 = 𝑀 → (∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏) ↔ ∀𝑏 ∈ 𝑀 (𝐺‘𝑀) ≠ (𝐺‘𝑏)))
16	15	imbi2d 340	. . . . 5 ⊢ (𝑐 = 𝑀 → ((𝜑 → ∀𝑏 ∈ 𝑐 (𝐺‘𝑐) ≠ (𝐺‘𝑏)) ↔ (𝜑 → ∀𝑏 ∈ 𝑀 (𝐺‘𝑀) ≠ (𝐺‘𝑏))))
17		ral0 4440	. . . . . 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 6712	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝐵–1-1-onto→𝐴 → ^◡𝐹:𝐴–1-1-onto→𝐵)
21		f1of 6700	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡𝐹:𝐴–1-1-onto→𝐵 → ^◡𝐹:𝐴⟶𝐵)
22	19, 20, 21	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ^◡𝐹:𝐴⟶𝐵)
23	22	adantl 481	. . . . . . . . . . . . . . . 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 9992	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺:ω⟶𝐴)
28	27	ffvelrnda 6943	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → (𝐺‘𝑑) ∈ 𝐴)
29	28	ancoms 458	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → (𝐺‘𝑑) ∈ 𝐴)
30	23, 29	ffvelrnd 6944	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → (^◡𝐹‘(𝐺‘𝑑)) ∈ 𝐵)
31	25	eldifbd 3896	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)
32	31	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → ¬ 𝐶 ∈ 𝐵)
33		nelne2 3041	. . . . . . . . . . . . . . 15 ⊢ (((^◡𝐹‘(𝐺‘𝑑)) ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → (^◡𝐹‘(𝐺‘𝑑)) ≠ 𝐶)
34	30, 32, 33	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → (^◡𝐹‘(𝐺‘𝑑)) ≠ 𝐶)
35	24, 19, 25, 26	infpssrlem2 9991	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ ω → (𝐺‘suc 𝑑) = (^◡𝐹‘(𝐺‘𝑑)))
36	35	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → (𝐺‘suc 𝑑) = (^◡𝐹‘(𝐺‘𝑑)))
37	24, 19, 25, 26	infpssrlem1 9990	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
38	37	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → (𝐺‘∅) = 𝐶)
39	34, 36, 38	3netr4d 3020	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ ω ∧ 𝜑) → (𝐺‘suc 𝑑) ≠ (𝐺‘∅))
40	39	3adant3 1130	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) → (𝐺‘suc 𝑑) ≠ (𝐺‘∅))
41	1	neeq2d 3003	. . . . . . . . . . . 12 ⊢ (𝑐 = ∅ → ((𝐺‘suc 𝑑) ≠ (𝐺‘𝑐) ↔ (𝐺‘suc 𝑑) ≠ (𝐺‘∅)))
42	40, 41	syl5ibr 245	. . . . . . . . . . 11 ⊢ (𝑐 = ∅ → ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐)))
43	42	adantrd 491	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑) → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐)))
44		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∈ ω ∧ 𝑐 ∈ suc 𝑑) → 𝑐 ∈ suc 𝑑)
45		peano2 7711	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ ω → suc 𝑑 ∈ ω)
46	45	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∈ ω ∧ 𝑐 ∈ suc 𝑑) → suc 𝑑 ∈ ω)
47		elnn 7698	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ suc 𝑑 ∧ suc 𝑑 ∈ ω) → 𝑐 ∈ ω)
48	44, 46, 47	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ ω ∧ 𝑐 ∈ suc 𝑑) → 𝑐 ∈ ω)
49	48	3ad2antl1 1183	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑) → 𝑐 ∈ ω)
50	49	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑐 ≠ ∅ ∧ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑)) → 𝑐 ∈ ω)
51		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑐 ≠ ∅ ∧ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑)) → 𝑐 ≠ ∅)
52		nnsuc 7705	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ ω ∧ 𝑐 ≠ ∅) → ∃𝑏 ∈ ω 𝑐 = suc 𝑏)
53	50, 51, 52	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝑐 ≠ ∅ ∧ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑)) → ∃𝑏 ∈ ω 𝑐 = suc 𝑏)
54		nfv 1918	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑏 𝑑 ∈ ω
55		nfv 1918	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑏𝜑
56		nfra1 3142	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑏∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)
57	54, 55, 56	nf3an 1905	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑏(𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏))
58		nfv 1918	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑏 𝑐 ∈ suc 𝑑
59	57, 58	nfan 1903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑)
60		nfv 1918	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏(𝐺‘suc 𝑑) ≠ (𝐺‘𝑐)
61		simpl3 1191	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ (suc 𝑏 ∈ suc 𝑑 ∧ 𝑏 ∈ ω)) → ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏))
62		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑑 ∈ ω ∧ suc 𝑏 ∈ suc 𝑑) → suc 𝑏 ∈ suc 𝑑)
63		nnord 7695	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑑 ∈ ω → Ord 𝑑)
64	63	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑑 ∈ ω ∧ suc 𝑏 ∈ suc 𝑑) → Ord 𝑑)
65		ordsucelsuc 7644	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Ord 𝑑 → (𝑏 ∈ 𝑑 ↔ suc 𝑏 ∈ suc 𝑑))
66	64, 65	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑑 ∈ ω ∧ suc 𝑏 ∈ suc 𝑑) → (𝑏 ∈ 𝑑 ↔ suc 𝑏 ∈ suc 𝑑))
67	62, 66	mpbird 256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑑 ∈ ω ∧ suc 𝑏 ∈ suc 𝑑) → 𝑏 ∈ 𝑑)
68	67	3ad2antl1 1183	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ suc 𝑏 ∈ suc 𝑑) → 𝑏 ∈ 𝑑)
69	68	adantrr 713	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ (suc 𝑏 ∈ suc 𝑑 ∧ 𝑏 ∈ ω)) → 𝑏 ∈ 𝑑)
70		rsp 3129	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏) → (𝑏 ∈ 𝑑 → (𝐺‘𝑑) ≠ (𝐺‘𝑏)))
71	61, 69, 70	sylc 65	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ (suc 𝑏 ∈ suc 𝑑 ∧ 𝑏 ∈ ω)) → (𝐺‘𝑑) ≠ (𝐺‘𝑏))
72		f1of1 6699	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (^◡𝐹:𝐴–1-1-onto→𝐵 → ^◡𝐹:𝐴–1-1→𝐵)
73	19, 20, 72	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ^◡𝐹:𝐴–1-1→𝐵)
74	73	ad2antlr 723	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → ^◡𝐹:𝐴–1-1→𝐵)
75	29	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → (𝐺‘𝑑) ∈ 𝐴)
76	27	ffvelrnda 6943	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑏 ∈ ω) → (𝐺‘𝑏) ∈ 𝐴)
77	76	adantll 710	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → (𝐺‘𝑏) ∈ 𝐴)
78		f1fveq 7116	. . . . . . . . . . . . . . . . . . . . . . . 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 833	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → ((^◡𝐹‘(𝐺‘𝑑)) = (^◡𝐹‘(𝐺‘𝑏)) ↔ (𝐺‘𝑑) = (𝐺‘𝑏)))
80	79	necon3bid 2987	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → ((^◡𝐹‘(𝐺‘𝑑)) ≠ (^◡𝐹‘(𝐺‘𝑏)) ↔ (𝐺‘𝑑) ≠ (𝐺‘𝑏)))
81	80	biimprd 247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → ((𝐺‘𝑑) ≠ (𝐺‘𝑏) → (^◡𝐹‘(𝐺‘𝑑)) ≠ (^◡𝐹‘(𝐺‘𝑏))))
82	35	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑑 ∈ ω ∧ 𝑏 ∈ ω) → (𝐺‘suc 𝑑) = (^◡𝐹‘(𝐺‘𝑑)))
83	24, 19, 25, 26	infpssrlem2 9991	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 ∈ ω → (𝐺‘suc 𝑏) = (^◡𝐹‘(𝐺‘𝑏)))
84	83	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑑 ∈ ω ∧ 𝑏 ∈ ω) → (𝐺‘suc 𝑏) = (^◡𝐹‘(𝐺‘𝑏)))
85	82, 84	neeq12d 3004	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑑 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏) ↔ (^◡𝐹‘(𝐺‘𝑑)) ≠ (^◡𝐹‘(𝐺‘𝑏))))
86	85	adantlr 711	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → ((𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏) ↔ (^◡𝐹‘(𝐺‘𝑑)) ≠ (^◡𝐹‘(𝐺‘𝑏))))
87	81, 86	sylibrd 258	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ 𝑏 ∈ ω) → ((𝐺‘𝑑) ≠ (𝐺‘𝑏) → (𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏)))
88	87	adantrl 712	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑑 ∈ ω ∧ 𝜑) ∧ (suc 𝑏 ∈ suc 𝑑 ∧ 𝑏 ∈ ω)) → ((𝐺‘𝑑) ≠ (𝐺‘𝑏) → (𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏)))
89	88	3adantl3 1166	. . . . . . . . . . . . . . . . . 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 15	. . . . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . . . 16 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ suc 𝑏 ∈ suc 𝑑) → (𝑏 ∈ ω → (𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏)))
92		eleq1 2826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = suc 𝑏 → (𝑐 ∈ suc 𝑑 ↔ suc 𝑏 ∈ suc 𝑑))
93	92	anbi2d 628	. . . . . . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = suc 𝑏 → (𝐺‘𝑐) = (𝐺‘suc 𝑏))
95	94	neeq2d 3003	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = suc 𝑏 → ((𝐺‘suc 𝑑) ≠ (𝐺‘𝑐) ↔ (𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏)))
96	95	imbi2d 340	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = suc 𝑏 → ((𝑏 ∈ ω → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐)) ↔ (𝑏 ∈ ω → (𝐺‘suc 𝑑) ≠ (𝐺‘suc 𝑏))))
97	93, 96	imbi12d 344	. . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . 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 89	. . . . . . . . . . . . . 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 3245	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 15	. . . . . . . . . . 11 ⊢ ((𝑐 ≠ ∅ ∧ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑)) → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐))
103	102	ex 412	. . . . . . . . . 10 ⊢ (𝑐 ≠ ∅ → (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑) → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐)))
104	43, 103	pm2.61ine 3027	. . . . . . . . 9 ⊢ (((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) ∧ 𝑐 ∈ suc 𝑑) → (𝐺‘suc 𝑑) ≠ (𝐺‘𝑐))
105	104	ralrimiva 3107	. . . . . . . 8 ⊢ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) → ∀𝑐 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑐))
106		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → (𝐺‘𝑐) = (𝐺‘𝑏))
107	106	neeq2d 3003	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → ((𝐺‘suc 𝑑) ≠ (𝐺‘𝑐) ↔ (𝐺‘suc 𝑑) ≠ (𝐺‘𝑏)))
108	107	cbvralvw 3372	. . . . . . . 8 ⊢ (∀𝑐 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑐) ↔ ∀𝑏 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑏))
109	105, 108	sylib 217	. . . . . . 7 ⊢ ((𝑑 ∈ ω ∧ 𝜑 ∧ ∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏)) → ∀𝑏 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑏))
110	109	3exp 1117	. . . . . 6 ⊢ (𝑑 ∈ ω → (𝜑 → (∀𝑏 ∈ 𝑑 (𝐺‘𝑑) ≠ (𝐺‘𝑏) → ∀𝑏 ∈ suc 𝑑(𝐺‘suc 𝑑) ≠ (𝐺‘𝑏))))
111	110	a2d 29	. . . . 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 7719	. . . 4 ⊢ (𝑀 ∈ ω → (𝜑 → ∀𝑏 ∈ 𝑀 (𝐺‘𝑀) ≠ (𝐺‘𝑏)))
113	112	impcom 407	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ ω) → ∀𝑏 ∈ 𝑀 (𝐺‘𝑀) ≠ (𝐺‘𝑏))
114		fveq2 6756	. . . . 5 ⊢ (𝑏 = 𝑁 → (𝐺‘𝑏) = (𝐺‘𝑁))
115	114	neeq2d 3003	. . . 4 ⊢ (𝑏 = 𝑁 → ((𝐺‘𝑀) ≠ (𝐺‘𝑏) ↔ (𝐺‘𝑀) ≠ (𝐺‘𝑁)))
116	115	rspccv 3549	. . 3 ⊢ (∀𝑏 ∈ 𝑀 (𝐺‘𝑀) ≠ (𝐺‘𝑏) → (𝑁 ∈ 𝑀 → (𝐺‘𝑀) ≠ (𝐺‘𝑁)))
117	113, 116	syl 17	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ ω) → (𝑁 ∈ 𝑀 → (𝐺‘𝑀) ≠ (𝐺‘𝑁)))
118	117	3impia 1115	1 ⊢ ((𝜑 ∧ 𝑀 ∈ ω ∧ 𝑁 ∈ 𝑀) → (𝐺‘𝑀) ≠ (𝐺‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 ∖ cdif 3880 ⊆ wss 3883 ∅c0 4253 ^◡ccnv 5579 ↾ cres 5582 Ord word 6250 suc csuc 6253 ⟶wf 6414 –1-1→wf1 6415 –1-1-onto→wf1o 6417 ‘cfv 6418 ωcom 7687 reccrdg 8211

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212

This theorem is referenced by: infpssrlem5 9994

W3C validator