Theorem tfrlem1 8014

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 3991	. 2 ⊢ 𝐴 ⊆ 𝐴
2		tfrlem1.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
3		sseq1 3994	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴))
4		raleq 3407	. . . . . 6 ⊢ (𝑦 = 𝑧 → (∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥)))
5	3, 4	imbi12d 347	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))))
6	5	imbi2d 343	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝜑 → (𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥))) ↔ (𝜑 → (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥)))))
7		sseq1 3994	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
8		raleq 3407	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
9	7, 8	imbi12d 347	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (𝐴 ⊆ 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
10	9	imbi2d 343	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝜑 → (𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥))) ↔ (𝜑 → (𝐴 ⊆ 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))))
11		r19.21v 3177	. . . . 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 730	. . . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → Fun 𝐹)
15	14	funfnd 6388	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝐹 Fn dom 𝐹)
16		eloni 6203	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ On → Ord 𝑦)
17	16	ad3antlr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) → Ord 𝑦)
18		ordelss 6209	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ⊆ 𝑦)
19	17, 18	sylan 582	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝑤 ⊆ 𝑦)
20		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝑦 ⊆ 𝐴)
21	19, 20	sstrd 3979	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝑤 ⊆ 𝐴)
22	13	simprd 498	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝐴 ⊆ dom 𝐹)
23	21, 22	sstrd 3979	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝑤 ⊆ dom 𝐹)
24		fnssres 6472	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝑤 ⊆ dom 𝐹) → (𝐹 ↾ 𝑤) Fn 𝑤)
25	15, 23, 24	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐹 ↾ 𝑤) Fn 𝑤)
26		tfrlem1.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Fun 𝐺 ∧ 𝐴 ⊆ dom 𝐺))
27	26	ad4antr 730	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (Fun 𝐺 ∧ 𝐴 ⊆ dom 𝐺))
28	27	simpld 497	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → Fun 𝐺)
29	28	funfnd 6388	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝐺 Fn dom 𝐺)
30	27	simprd 498	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝐴 ⊆ dom 𝐺)
31	21, 30	sstrd 3979	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝑤 ⊆ dom 𝐺)
32		fnssres 6472	. . . . . . . . . . . . . 14 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝑤 ⊆ dom 𝐺) → (𝐺 ↾ 𝑤) Fn 𝑤)
33	29, 31, 32	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐺 ↾ 𝑤) Fn 𝑤)
34		fveq2 6672	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑢 → (𝐹‘𝑥) = (𝐹‘𝑢))
35		fveq2 6672	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑢 → (𝐺‘𝑥) = (𝐺‘𝑢))
36	34, 35	eqeq12d 2839	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝑢) = (𝐺‘𝑢)))
37	21	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → 𝑤 ⊆ 𝐴)
38		sseq1 3994	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (𝑧 ⊆ 𝐴 ↔ 𝑤 ⊆ 𝐴))
39		raleq 3407	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑤 (𝐹‘𝑥) = (𝐺‘𝑥)))
40	38, 39	imbi12d 347	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → ((𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (𝑤 ⊆ 𝐴 → ∀𝑥 ∈ 𝑤 (𝐹‘𝑥) = (𝐺‘𝑥))))
41		simp-4r 782	. . . . . . . . . . . . . . . . 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)))
42		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → 𝑤 ∈ 𝑦)
43	40, 41, 42	rspcdva 3627	. . . . . . . . . . . . . . . 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)))
44	37, 43	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))
45		simpr 487	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → 𝑢 ∈ 𝑤)
46	36, 44, 45	rspcdva 3627	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) ∧ 𝑢 ∈ 𝑤) → (𝐹‘𝑢) = (𝐺‘𝑢))
47		fvres 6691	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑤 → ((𝐹 ↾ 𝑤)‘𝑢) = (𝐹‘𝑢))
48	47	adantl 484	. . . . . . . . . . . . . 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))
49		fvres 6691	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑤 → ((𝐺 ↾ 𝑤)‘𝑢) = (𝐺‘𝑢))
50	49	adantl 484	. . . . . . . . . . . . . 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))
51	46, 48, 50	3eqtr4d 2868	. . . . . . . . . . . . 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))
52	25, 33, 51	eqfnfvd 6807	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐹 ↾ 𝑤) = (𝐺 ↾ 𝑤))
53	52	fveq2d 6676	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐵‘(𝐹 ↾ 𝑤)) = (𝐵‘(𝐺 ↾ 𝑤)))
54		fveq2 6672	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
55		reseq2 5850	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝑤))
56	55	fveq2d 6676	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝐵‘(𝐹 ↾ 𝑥)) = (𝐵‘(𝐹 ↾ 𝑤)))
57	54, 56	eqeq12d 2839	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) = (𝐵‘(𝐹 ↾ 𝑥)) ↔ (𝐹‘𝑤) = (𝐵‘(𝐹 ↾ 𝑤))))
58		tfrlem1.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐵‘(𝐹 ↾ 𝑥)))
59	58	ad4antr 730	. . . . . . . . . . . 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)))
60		simpr 487	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
61	60	sselda 3969	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝐴)
62	57, 59, 61	rspcdva 3627	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐹‘𝑤) = (𝐵‘(𝐹 ↾ 𝑤)))
63		fveq2 6672	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝐺‘𝑥) = (𝐺‘𝑤))
64		reseq2 5850	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝐺 ↾ 𝑥) = (𝐺 ↾ 𝑤))
65	64	fveq2d 6676	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝐵‘(𝐺 ↾ 𝑥)) = (𝐵‘(𝐺 ↾ 𝑤)))
66	63, 65	eqeq12d 2839	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝐺‘𝑥) = (𝐵‘(𝐺 ↾ 𝑥)) ↔ (𝐺‘𝑤) = (𝐵‘(𝐺 ↾ 𝑤))))
67		tfrlem1.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) = (𝐵‘(𝐺 ↾ 𝑥)))
68	67	ad4antr 730	. . . . . . . . . . . 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)))
69	66, 68, 61	rspcdva 3627	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐺‘𝑤) = (𝐵‘(𝐺 ↾ 𝑤)))
70	53, 62, 69	3eqtr4d 2868	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝐹‘𝑤) = (𝐺‘𝑤))
71	70	ralrimiva 3184	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) → ∀𝑤 ∈ 𝑦 (𝐹‘𝑤) = (𝐺‘𝑤))
72	54, 63	eqeq12d 2839	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝑤) = (𝐺‘𝑤)))
73	72	cbvralvw 3451	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥) ↔ ∀𝑤 ∈ 𝑦 (𝐹‘𝑤) = (𝐺‘𝑤))
74	71, 73	sylibr 236	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ On) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ⊆ 𝐴) → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥))
75	74	exp31 422	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥)) → (𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥))))
76	75	expcom 416	. . . . . 6 ⊢ (𝑦 ∈ On → (𝜑 → (∀𝑧 ∈ 𝑦 (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥)) → (𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥)))))
77	76	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)))))
78	11, 77	syl5bi 244	. . . 4 ⊢ (𝑦 ∈ On → (∀𝑧 ∈ 𝑦 (𝜑 → (𝑧 ⊆ 𝐴 → ∀𝑥 ∈ 𝑧 (𝐹‘𝑥) = (𝐺‘𝑥))) → (𝜑 → (𝑦 ⊆ 𝐴 → ∀𝑥 ∈ 𝑦 (𝐹‘𝑥) = (𝐺‘𝑥)))))
79	6, 10, 78	tfis3 7574	. . 3 ⊢ (𝐴 ∈ On → (𝜑 → (𝐴 ⊆ 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
80	2, 79	mpcom 38	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
81	1, 80	mpi 20	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140   ⊆ wss 3938  dom cdm 5557   ↾ cres 5559  Ord word 6192  Oncon0 6193  Fun wfun 6351   Fn wfn 6352  ‘cfv 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-iota 6316  df-fun 6359  df-fn 6360  df-fv 6365

This theorem is referenced by:  tfrlem5  8018