Theorem tfrlem1 7625

Description: A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by Mario Carneiro, 24-May-2019.)

Hypotheses

Ref	Expression
tfrlem1.1	⊢ (𝜑 → 𝐴 ∈ On)
tfrlem1.2	⊢ (𝜑 → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹))
tfrlem1.3	⊢ (𝜑 → (Fun 𝐺 ∧ 𝐴 ⊆ dom 𝐺))
tfrlem1.4	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐵‘(𝐹 ↾ 𝑥)))
tfrlem1.5	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) = (𝐵‘(𝐺 ↾ 𝑥)))

Assertion

Ref	Expression
tfrlem1	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem tfrlem1

Dummy variables 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3773	. 2 ⊢ 𝐴 ⊆ 𝐴
2		tfrlem1.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
3		sseq1 3775	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴))
4		raleq 3287	. . . . . 6 ⊢ (𝑦 = 𝑧 → (∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥)))
5	3, 4	imbi12d 333	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))))
6	5	imbi2d 329	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝜑 → (𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥))) ↔ (𝜑 → (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥)))))
7		sseq1 3775	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
8		raleq 3287	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
9	7, 8	imbi12d 333	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (𝐴 ⊆ 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
10	9	imbi2d 329	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝜑 → (𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥))) ↔ (𝜑 → (𝐴 ⊆ 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))))
11		r19.21v 3109	. . . . 5 ⊢ (∀𝑧 ∈ 𝑦 (𝜑 → (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ↔ (𝜑 → ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))))
12		tfrlem1.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹))
13	12	ad4antr 712	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹))
14	13	simpld 482	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → Fun 𝐹)
15		funfn 6061	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
16	14, 15	sylib 208	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝐹 Fn dom 𝐹)
17		eloni 5876	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ On → Ord 𝑦)
18	17	ad3antlr 710	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) → Ord 𝑦)
19		ordelss 5882	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ⊆ 𝑦)
20	18, 19	sylan 569	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝑤 ⊆ 𝑦)
21		simplr 752	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝑦 ⊆ 𝐴)
22	20, 21	sstrd 3762	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝑤 ⊆ 𝐴)
23	13	simprd 483	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝐴 ⊆ dom 𝐹)
24	22, 23	sstrd 3762	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝑤 ⊆ dom 𝐹)
25		fnssres 6144	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝑤 ⊆ dom 𝐹) → (𝐹 ↾ 𝑤) Fn 𝑤)
26	16, 24, 25	syl2anc 573	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐹 ↾ 𝑤) Fn 𝑤)
27		tfrlem1.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Fun 𝐺 ∧ 𝐴 ⊆ dom 𝐺))
28	27	ad4antr 712	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (Fun 𝐺 ∧ 𝐴 ⊆ dom 𝐺))
29	28	simpld 482	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → Fun 𝐺)
30		funfn 6061	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺)
31	29, 30	sylib 208	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝐺 Fn dom 𝐺)
32	28	simprd 483	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝐴 ⊆ dom 𝐺)
33	22, 32	sstrd 3762	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝑤 ⊆ dom 𝐺)
34		fnssres 6144	. . . . . . . . . . . . . 14 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝑤 ⊆ dom 𝐺) → (𝐺 ↾ 𝑤) Fn 𝑤)
35	31, 33, 34	syl2anc 573	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐺 ↾ 𝑤) Fn 𝑤)
36		fveq2 6332	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑢 → (𝐹‘𝑥) = (𝐹‘𝑢))
37		fveq2 6332	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑢 → (𝐺‘𝑥) = (𝐺‘𝑢))
38	36, 37	eqeq12d 2786	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝑢) = (𝐺‘𝑢)))
39	22	adantr 466	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → 𝑤 ⊆ 𝐴)
40		sseq1 3775	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (𝑧 ⊆ 𝐴 ↔ 𝑤 ⊆ 𝐴))
41		raleq 3287	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑤 (𝐹‘𝑥) = (𝐺‘𝑥)))
42	40, 41	imbi12d 333	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → ((𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (𝑤 ⊆ 𝐴 → ∀𝑥 ∈ 𝑤 (𝐹‘𝑥) = (𝐺‘𝑥))))
43		simp-4r 770	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥)))
44		simplr 752	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → 𝑤 ∈ 𝑦)
45	42, 43, 44	rspcdva 3466	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → (𝑤 ⊆ 𝐴 → ∀𝑥 ∈ 𝑤 (𝐹‘𝑥) = (𝐺‘𝑥)))
46	39, 45	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → ∀𝑥 ∈ 𝑤 (𝐹‘𝑥) = (𝐺‘𝑥))
47		simpr 471	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → 𝑢 ∈ 𝑤)
48	38, 46, 47	rspcdva 3466	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → (𝐹‘𝑢) = (𝐺‘𝑢))
49		fvres 6348	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑤 → ((𝐹 ↾ 𝑤)‘𝑢) = (𝐹‘𝑢))
50	49	adantl 467	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → ((𝐹 ↾ 𝑤)‘𝑢) = (𝐹‘𝑢))
51		fvres 6348	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑤 → ((𝐺 ↾ 𝑤)‘𝑢) = (𝐺‘𝑢))
52	51	adantl 467	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → ((𝐺 ↾ 𝑤)‘𝑢) = (𝐺‘𝑢))
53	48, 50, 52	3eqtr4d 2815	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → ((𝐹 ↾ 𝑤)‘𝑢) = ((𝐺 ↾ 𝑤)‘𝑢))
54	26, 35, 53	eqfnfvd 6457	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐹 ↾ 𝑤) = (𝐺 ↾ 𝑤))
55	54	fveq2d 6336	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐵‘(𝐹 ↾ 𝑤)) = (𝐵‘(𝐺 ↾ 𝑤)))
56		fveq2 6332	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
57		reseq2 5529	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝑤))
58	57	fveq2d 6336	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝐵‘(𝐹 ↾ 𝑥)) = (𝐵‘(𝐹 ↾ 𝑤)))
59	56, 58	eqeq12d 2786	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) = (𝐵‘(𝐹 ↾ 𝑥)) ↔ (𝐹‘𝑤) = (𝐵‘(𝐹 ↾ 𝑤))))
60		tfrlem1.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐵‘(𝐹 ↾ 𝑥)))
61	60	ad4antr 712	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐵‘(𝐹 ↾ 𝑥)))
62		simpr 471	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
63	62	sselda 3752	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝐴)
64	59, 61, 63	rspcdva 3466	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐹‘𝑤) = (𝐵‘(𝐹 ↾ 𝑤)))
65		fveq2 6332	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝐺‘𝑥) = (𝐺‘𝑤))
66		reseq2 5529	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝐺 ↾ 𝑥) = (𝐺 ↾ 𝑤))
67	66	fveq2d 6336	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝐵‘(𝐺 ↾ 𝑥)) = (𝐵‘(𝐺 ↾ 𝑤)))
68	65, 67	eqeq12d 2786	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝐺‘𝑥) = (𝐵‘(𝐺 ↾ 𝑥)) ↔ (𝐺‘𝑤) = (𝐵‘(𝐺 ↾ 𝑤))))
69		tfrlem1.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) = (𝐵‘(𝐺 ↾ 𝑥)))
70	69	ad4antr 712	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) = (𝐵‘(𝐺 ↾ 𝑥)))
71	68, 70, 63	rspcdva 3466	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐺‘𝑤) = (𝐵‘(𝐺 ↾ 𝑤)))
72	55, 64, 71	3eqtr4d 2815	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐹‘𝑤) = (𝐺‘𝑤))
73	72	ralrimiva 3115	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) → ∀𝑤 ∈ 𝑦 (𝐹‘𝑤) = (𝐺‘𝑤))
74	56, 65	eqeq12d 2786	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝑤) = (𝐺‘𝑤)))
75	74	cbvralv 3320	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥) ↔ ∀𝑤 ∈ 𝑦 (𝐹‘𝑤) = (𝐺‘𝑤))
76	73, 75	sylibr 224	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥))
77	76	exp31 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥)) → (𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥))))
78	77	expcom 398	. . . . . 6 ⊢ (𝑦 ∈ On → (𝜑 → (∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥)) → (𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥)))))
79	78	a2d 29	. . . . 5 ⊢ (𝑦 ∈ On → ((𝜑 → ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) → (𝜑 → (𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥)))))
80	11, 79	syl5bi 232	. . . 4 ⊢ (𝑦 ∈ On → (∀𝑧 ∈ 𝑦 (𝜑 → (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) → (𝜑 → (𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥)))))
81	6, 10, 80	tfis3 7204	. . 3 ⊢ (𝐴 ∈ On → (𝜑 → (𝐴 ⊆ 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
82	2, 81	mpcom 38	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
83	1, 82	mpi 20	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061   ⊆ wss 3723  dom cdm 5249   ↾ cres 5251  Ord word 5865  Oncon0 5866  Fun wfun 6025   Fn wfn 6026  ‘cfv 6031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039

This theorem is referenced by:  tfrlem5  7629