Theorem tfrlem5 8019

Description: Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)

Hypothesis

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Assertion

Ref	Expression
tfrlem5	⊢ ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))

Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,ℎ,𝑢,𝑣,𝐹   𝐴,𝑔,ℎ

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑣,𝑢,𝑓)

Proof of Theorem tfrlem5

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

Step	Hyp	Ref	Expression
1		tfrlem.1	. . 3 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2		vex 3500	. . 3 ⊢ 𝑔 ∈ V
3	1, 2	tfrlem3a 8016	. 2 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))))
4		vex 3500	. . 3 ⊢ ℎ ∈ V
5	1, 4	tfrlem3a 8016	. 2 ⊢ (ℎ ∈ 𝐴 ↔ ∃𝑤 ∈ On (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎))))
6		reeanv 3370	. . 3 ⊢ (∃𝑧 ∈ On ∃𝑤 ∈ On ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ↔ (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ ∃𝑤 ∈ On (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))))
7		fveq2 6673	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑔‘𝑎) = (𝑔‘𝑥))
8		fveq2 6673	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (ℎ‘𝑎) = (ℎ‘𝑥))
9	7, 8	eqeq12d 2840	. . . . . . 7 ⊢ (𝑎 = 𝑥 → ((𝑔‘𝑎) = (ℎ‘𝑎) ↔ (𝑔‘𝑥) = (ℎ‘𝑥)))
10		onin 6225	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 ∩ 𝑤) ∈ On)
11	10	3ad2ant1 1129	. . . . . . . 8 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → (𝑧 ∩ 𝑤) ∈ On)
12		simp2ll 1236	. . . . . . . . . 10 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → 𝑔 Fn 𝑧)
13		fnfun 6456	. . . . . . . . . 10 ⊢ (𝑔 Fn 𝑧 → Fun 𝑔)
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → Fun 𝑔)
15		inss1 4208	. . . . . . . . . 10 ⊢ (𝑧 ∩ 𝑤) ⊆ 𝑧
16		fndm 6458	. . . . . . . . . . 11 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
17	12, 16	syl 17	. . . . . . . . . 10 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → dom 𝑔 = 𝑧)
18	15, 17	sseqtrrid 4023	. . . . . . . . 9 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → (𝑧 ∩ 𝑤) ⊆ dom 𝑔)
19	14, 18	jca 514	. . . . . . . 8 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → (Fun 𝑔 ∧ (𝑧 ∩ 𝑤) ⊆ dom 𝑔))
20		simp2rl 1238	. . . . . . . . . 10 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → ℎ Fn 𝑤)
21		fnfun 6456	. . . . . . . . . 10 ⊢ (ℎ Fn 𝑤 → Fun ℎ)
22	20, 21	syl 17	. . . . . . . . 9 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → Fun ℎ)
23		inss2 4209	. . . . . . . . . 10 ⊢ (𝑧 ∩ 𝑤) ⊆ 𝑤
24		fndm 6458	. . . . . . . . . . 11 ⊢ (ℎ Fn 𝑤 → dom ℎ = 𝑤)
25	20, 24	syl 17	. . . . . . . . . 10 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → dom ℎ = 𝑤)
26	23, 25	sseqtrrid 4023	. . . . . . . . 9 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → (𝑧 ∩ 𝑤) ⊆ dom ℎ)
27	22, 26	jca 514	. . . . . . . 8 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → (Fun ℎ ∧ (𝑧 ∩ 𝑤) ⊆ dom ℎ))
28		simp2lr 1237	. . . . . . . . 9 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎)))
29		ssralv 4036	. . . . . . . . 9 ⊢ ((𝑧 ∩ 𝑤) ⊆ 𝑧 → (∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎)) → ∀𝑎 ∈ (𝑧 ∩ 𝑤)(𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))))
30	15, 28, 29	mpsyl 68	. . . . . . . 8 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → ∀𝑎 ∈ (𝑧 ∩ 𝑤)(𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎)))
31		simp2rr 1239	. . . . . . . . 9 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))
32		ssralv 4036	. . . . . . . . 9 ⊢ ((𝑧 ∩ 𝑤) ⊆ 𝑤 → (∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)) → ∀𝑎 ∈ (𝑧 ∩ 𝑤)(ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎))))
33	23, 31, 32	mpsyl 68	. . . . . . . 8 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → ∀𝑎 ∈ (𝑧 ∩ 𝑤)(ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))
34	11, 19, 27, 30, 33	tfrlem1 8015	. . . . . . 7 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → ∀𝑎 ∈ (𝑧 ∩ 𝑤)(𝑔‘𝑎) = (ℎ‘𝑎))
35		simp3l 1197	. . . . . . . . 9 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → 𝑥𝑔𝑢)
36		fnbr 6462	. . . . . . . . 9 ⊢ ((𝑔 Fn 𝑧 ∧ 𝑥𝑔𝑢) → 𝑥 ∈ 𝑧)
37	12, 35, 36	syl2anc 586	. . . . . . . 8 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → 𝑥 ∈ 𝑧)
38		simp3r 1198	. . . . . . . . 9 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → 𝑥ℎ𝑣)
39		fnbr 6462	. . . . . . . . 9 ⊢ ((ℎ Fn 𝑤 ∧ 𝑥ℎ𝑣) → 𝑥 ∈ 𝑤)
40	20, 38, 39	syl2anc 586	. . . . . . . 8 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → 𝑥 ∈ 𝑤)
41	37, 40	elind 4174	. . . . . . 7 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → 𝑥 ∈ (𝑧 ∩ 𝑤))
42	9, 34, 41	rspcdva 3628	. . . . . 6 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → (𝑔‘𝑥) = (ℎ‘𝑥))
43		funbrfv 6719	. . . . . . 7 ⊢ (Fun 𝑔 → (𝑥𝑔𝑢 → (𝑔‘𝑥) = 𝑢))
44	14, 35, 43	sylc 65	. . . . . 6 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → (𝑔‘𝑥) = 𝑢)
45		funbrfv 6719	. . . . . . 7 ⊢ (Fun ℎ → (𝑥ℎ𝑣 → (ℎ‘𝑥) = 𝑣))
46	22, 38, 45	sylc 65	. . . . . 6 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → (ℎ‘𝑥) = 𝑣)
47	42, 44, 46	3eqtr3d 2867	. . . . 5 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) → 𝑢 = 𝑣)
48	47	3exp 1115	. . . 4 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)))
49	48	rexlimivv 3295	. . 3 ⊢ (∃𝑧 ∈ On ∃𝑤 ∈ On ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
50	6, 49	sylbir 237	. 2 ⊢ ((∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) ∧ ∃𝑤 ∈ On (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
51	3, 5, 50	syl2anb 599	1 ⊢ ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  {cab 2802  ∀wral 3141  ∃wrex 3142   ∩ cin 3938   ⊆ wss 3939   class class class wbr 5069  dom cdm 5558   ↾ cres 5560  Oncon0 6194  Fun wfun 6352   Fn wfn 6353  ‘cfv 6358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-ord 6197  df-on 6198  df-iota 6317  df-fun 6360  df-fn 6361  df-fv 6366

This theorem is referenced by:  tfrlem7  8022