Theorem tfrlem5 8369

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 3454	. . 3 ⊢ 𝑔 ∈ V
3	1, 2	tfrlem3a 8366	. 2 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))))
4		vex 3454	. . 3 ⊢ ℎ ∈ V
5	1, 4	tfrlem3a 8366	. 2 ⊢ (ℎ ∈ 𝐴 ↔ ∃𝑤 ∈ On (ℎ Fn 𝑤 ∧ ∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎))))
6		reeanv 3234	. . 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 6879	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑔‘𝑎) = (𝑔‘𝑥))
8		fveq2 6879	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (ℎ‘𝑎) = (ℎ‘𝑥))
9	7, 8	eqeq12d 2776	. . . . . . 7 ⊢ (𝑎 = 𝑥 → ((𝑔‘𝑎) = (ℎ‘𝑎) ↔ (𝑔‘𝑥) = (ℎ‘𝑥)))
10		onin 6389	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 ∩ 𝑤) ∈ On)
11	10	3ad2ant1 1151	. . . . . . . 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 1259	. . . . . . . . . 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 6633	. . . . . . . . . 10 ⊢ (𝑔 Fn 𝑧 → Fun 𝑔)
14	12, 13	syl 18	. . . . . . . . 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 4182	. . . . . . . . . 10 ⊢ (𝑧 ∩ 𝑤) ⊆ 𝑧
16	12	fndmd 6638	. . . . . . . . . 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)
17	15, 16	sseqtrrid 3974	. . . . . . . . 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)
18	14, 17	jca 521	. . . . . . . 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))
19		simp2rl 1261	. . . . . . . . . 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)
20		fnfun 6633	. . . . . . . . . 10 ⊢ (ℎ Fn 𝑤 → Fun ℎ)
21	19, 20	syl 18	. . . . . . . . 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)
22		inss2 4183	. . . . . . . . . 10 ⊢ (𝑧 ∩ 𝑤) ⊆ 𝑤
23	19	fndmd 6638	. . . . . . . . . 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)
24	22, 23	sseqtrrid 3974	. . . . . . . . 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)
25	21, 24	jca 521	. . . . . . . 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))
26		simp2lr 1260	. . . . . . . . 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)))
27		ssralv 4000	. . . . . . . . 9 ⊢ ((𝑧 ∩ 𝑤) ⊆ 𝑧 → (∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎)) → ∀𝑎 ∈ (𝑧 ∩ 𝑤)(𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))))
28	15, 26, 27	mpsyl 69	. . . . . . . 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)))
29		simp2rr 1262	. . . . . . . . 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)))
30		ssralv 4000	. . . . . . . . 9 ⊢ ((𝑧 ∩ 𝑤) ⊆ 𝑤 → (∀𝑎 ∈ 𝑤 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)) → ∀𝑎 ∈ (𝑧 ∩ 𝑤)(ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎))))
31	22, 29, 30	mpsyl 69	. . . . . . . 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)))
32	11, 18, 25, 28, 31	tfrlem1 8365	. . . . . . 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))
33		simp3l 1220	. . . . . . . . 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)
34		fnbr 6641	. . . . . . . . 9 ⊢ ((𝑔 Fn 𝑧 ∧ 𝑥𝑔𝑢) → 𝑥 ∈ 𝑧)
35	12, 33, 34	syl2anc 596	. . . . . . . 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)
36		simp3r 1221	. . . . . . . . 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)
37		fnbr 6641	. . . . . . . . 9 ⊢ ((ℎ Fn 𝑤 ∧ 𝑥ℎ𝑣) → 𝑥 ∈ 𝑤)
38	19, 36, 37	syl2anc 596	. . . . . . . 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)
39	35, 38	elind 4146	. . . . . . 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))
40	9, 32, 39	rspcdva 3577	. . . . . 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))
41		funbrfv 6927	. . . . . . 7 ⊢ (Fun 𝑔 → (𝑥𝑔𝑢 → (𝑔‘𝑥) = 𝑢))
42	14, 33, 41	sylc 66	. . . . . 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)
43		funbrfv 6927	. . . . . . 7 ⊢ (Fun ℎ → (𝑥ℎ𝑣 → (ℎ‘𝑥) = 𝑣))
44	21, 36, 43	sylc 66	. . . . . 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)
45	40, 42, 44	3eqtr3d 2803	. . . . 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)
46	45	3exp 1137	. . . 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)))
47	46	rexlimivv 3204	. . 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))
48	6, 47	sylbir 238	. 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))
49	3, 5, 48	syl2anb 610	1 ⊢ ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086   ∩ cin 3898   ⊆ wss 3899   class class class wbr 5103  dom cdm 5655   ↾ cres 5657  Oncon0 6357  Fun wfun 6527   Fn wfn 6528  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6360  df-on 6361  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541

This theorem is used by:  tfrlem7  8373