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Theorem jf1 35961

Description: 𝐽 is a one-to-one function. Part of Proposition 15.3 of [TakeutiZaring] p. 155. (Contributed by BTernaryTau, 6-Oct-2026.)

**Assertion**
Ref	Expression
jf1	⊢ 𝐽:((On × On) × 9_o)–1-1→On

Proof of Theorem jf1 Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		9on 35843	. . . . 5 ⊢ 9_o ∈ On
2		j0iso 35955	. . . . . . . 8 ⊢ 𝐽₀ Isom 𝑅₀, E ((On × On), On)
3		isof1o 7335	. . . . . . . 8 ⊢ (𝐽₀ Isom 𝑅₀, E ((On × On), On) → 𝐽₀:(On × On)–1-1-onto→On)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ 𝐽₀:(On × On)–1-1-onto→On
5		f1of 6829	. . . . . . 7 ⊢ (𝐽₀:(On × On)–1-1-onto→On → 𝐽₀:(On × On)⟶On)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ 𝐽₀:(On × On)⟶On
7		0elon 6421	. . . . . 6 ⊢ ∅ ∈ On
8	6, 7	f0cli 7103	. . . . 5 ⊢ (𝐽₀‘(1^st ‘𝑤)) ∈ On
9		omcl 8552	. . . . 5 ⊢ ((9_o ∈ On ∧ (𝐽₀‘(1^st ‘𝑤)) ∈ On) → (9_o ·_o (𝐽₀‘(1^st ‘𝑤))) ∈ On)
10	1, 8, 9	mp2an 705	. . . 4 ⊢ (9_o ·_o (𝐽₀‘(1^st ‘𝑤))) ∈ On
11		xp2nd 8038	. . . . 5 ⊢ (𝑤 ∈ ((On × On) × 9_o) → (2^nd ‘𝑤) ∈ 9_o)
12		onelon 6390	. . . . 5 ⊢ ((9_o ∈ On ∧ (2^nd ‘𝑤) ∈ 9_o) → (2^nd ‘𝑤) ∈ On)
13	1, 11, 12	sylancr 599	. . . 4 ⊢ (𝑤 ∈ ((On × On) × 9_o) → (2^nd ‘𝑤) ∈ On)
14		oacl 8551	. . . 4 ⊢ (((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) ∈ On ∧ (2^nd ‘𝑤) ∈ On) → ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) ∈ On)
15	10, 13, 14	sylancr 599	. . 3 ⊢ (𝑤 ∈ ((On × On) × 9_o) → ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) ∈ On)
16	15	rgen 3078	. 2 ⊢ ∀𝑤 ∈ ((On × On) × 9_o)((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) ∈ On
17	1	a1i 11	. . . . . . 7 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o) ∧ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧))) → 9_o ∈ On)
18		df-9o 35838	. . . . . . . . 9 ⊢ 9_o = suc 8_o
19		nsuceq0 6451	. . . . . . . . 9 ⊢ suc 8_o ≠ ∅
20	18, 19	eqnetri 3025	. . . . . . . 8 ⊢ 9_o ≠ ∅
21	20	a1i 11	. . . . . . 7 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o) ∧ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧))) → 9_o ≠ ∅)
22	8	a1i 11	. . . . . . 7 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o) ∧ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧))) → (𝐽₀‘(1^st ‘𝑤)) ∈ On)
23	11	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o) ∧ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧))) → (2^nd ‘𝑤) ∈ 9_o)
24	6, 7	f0cli 7103	. . . . . . . 8 ⊢ (𝐽₀‘(1^st ‘𝑧)) ∈ On
25	24	a1i 11	. . . . . . 7 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o) ∧ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧))) → (𝐽₀‘(1^st ‘𝑧)) ∈ On)
26		xp2nd 8038	. . . . . . . 8 ⊢ (𝑧 ∈ ((On × On) × 9_o) → (2^nd ‘𝑧) ∈ 9_o)
27	26	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o) ∧ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧))) → (2^nd ‘𝑧) ∈ 9_o)
28		simp3 1156	. . . . . . 7 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o) ∧ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧))) → ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧)))
29	17, 21, 22, 23, 25, 27, 28	omeueq 35773	. . . . . 6 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o) ∧ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧))) → ((𝐽₀‘(1^st ‘𝑤)) = (𝐽₀‘(1^st ‘𝑧)) ∧ (2^nd ‘𝑤) = (2^nd ‘𝑧)))
30		xp1st 8037	. . . . . . . . 9 ⊢ (𝑤 ∈ ((On × On) × 9_o) → (1^st ‘𝑤) ∈ (On × On))
31		xp1st 8037	. . . . . . . . 9 ⊢ (𝑧 ∈ ((On × On) × 9_o) → (1^st ‘𝑧) ∈ (On × On))
32		f1of1 6828	. . . . . . . . . . 11 ⊢ (𝐽₀:(On × On)–1-1-onto→On → 𝐽₀:(On × On)–1-1→On)
33	4, 32	ax-mp 5	. . . . . . . . . 10 ⊢ 𝐽₀:(On × On)–1-1→On
34		f1veqaeq 7264	. . . . . . . . . 10 ⊢ ((𝐽₀:(On × On)–1-1→On ∧ ((1^st ‘𝑤) ∈ (On × On) ∧ (1^st ‘𝑧) ∈ (On × On))) → ((𝐽₀‘(1^st ‘𝑤)) = (𝐽₀‘(1^st ‘𝑧)) → (1^st ‘𝑤) = (1^st ‘𝑧)))
35	33, 34	mpan 703	. . . . . . . . 9 ⊢ (((1^st ‘𝑤) ∈ (On × On) ∧ (1^st ‘𝑧) ∈ (On × On)) → ((𝐽₀‘(1^st ‘𝑤)) = (𝐽₀‘(1^st ‘𝑧)) → (1^st ‘𝑤) = (1^st ‘𝑧)))
36	30, 31, 35	syl2an 608	. . . . . . . 8 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o)) → ((𝐽₀‘(1^st ‘𝑤)) = (𝐽₀‘(1^st ‘𝑧)) → (1^st ‘𝑤) = (1^st ‘𝑧)))
37	36	anim1d 623	. . . . . . 7 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o)) → (((𝐽₀‘(1^st ‘𝑤)) = (𝐽₀‘(1^st ‘𝑧)) ∧ (2^nd ‘𝑤) = (2^nd ‘𝑧)) → ((1^st ‘𝑤) = (1^st ‘𝑧) ∧ (2^nd ‘𝑤) = (2^nd ‘𝑧))))
38	37	3adant3 1150	. . . . . 6 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o) ∧ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧))) → (((𝐽₀‘(1^st ‘𝑤)) = (𝐽₀‘(1^st ‘𝑧)) ∧ (2^nd ‘𝑤) = (2^nd ‘𝑧)) → ((1^st ‘𝑤) = (1^st ‘𝑧) ∧ (2^nd ‘𝑤) = (2^nd ‘𝑧))))
39	29, 38	mpd 16	. . . . 5 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o) ∧ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧))) → ((1^st ‘𝑤) = (1^st ‘𝑧) ∧ (2^nd ‘𝑤) = (2^nd ‘𝑧)))
40		xpopth 8046	. . . . . 6 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o)) → (((1^st ‘𝑤) = (1^st ‘𝑧) ∧ (2^nd ‘𝑤) = (2^nd ‘𝑧)) ↔ 𝑤 = 𝑧))
41	40	3adant3 1150	. . . . 5 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o) ∧ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧))) → (((1^st ‘𝑤) = (1^st ‘𝑧) ∧ (2^nd ‘𝑤) = (2^nd ‘𝑧)) ↔ 𝑤 = 𝑧))
42	39, 41	mpbid 235	. . . 4 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o) ∧ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧))) → 𝑤 = 𝑧)
43	42	3expia 1139	. . 3 ⊢ ((𝑤 ∈ ((On × On) × 9_o) ∧ 𝑧 ∈ ((On × On) × 9_o)) → (((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧)) → 𝑤 = 𝑧))
44	43	rgen2 3202	. 2 ⊢ ∀𝑤 ∈ ((On × On) × 9_o)∀𝑧 ∈ ((On × On) × 9_o)(((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧)) → 𝑤 = 𝑧)
45		dfj2 35960	. . 3 ⊢ 𝐽 = (𝑤 ∈ ((On × On) × 9_o) ↦ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)))
46		2fveq3 6895	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝐽₀‘(1^st ‘𝑤)) = (𝐽₀‘(1^st ‘𝑧)))
47	46	oveq2d 7440	. . . 4 ⊢ (𝑤 = 𝑧 → (9_o ·_o (𝐽₀‘(1^st ‘𝑤))) = (9_o ·_o (𝐽₀‘(1^st ‘𝑧))))
48		fveq2 6890	. . . 4 ⊢ (𝑤 = 𝑧 → (2^nd ‘𝑤) = (2^nd ‘𝑧))
49	47, 48	oveq12d 7442	. . 3 ⊢ (𝑤 = 𝑧 → ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧)))
50	45, 49	f1mpt 7269	. 2 ⊢ (𝐽:((On × On) × 9_o)–1-1→On ↔ (∀𝑤 ∈ ((On × On) × 9_o)((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) ∈ On ∧ ∀𝑤 ∈ ((On × On) × 9_o)∀𝑧 ∈ ((On × On) × 9_o)(((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)) = ((9_o ·_o (𝐽₀‘(1^st ‘𝑧))) +_o (2^nd ‘𝑧)) → 𝑤 = 𝑧)))
51	16, 44, 50	mpbir2an 724	1 ⊢ 𝐽:((On × On) × 9_o)–1-1→On

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∅c0 4277 E cep 5550 × cxp 5649 Oncon0 6365 suc csuc 6367 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 Isom wiso 6544 (class class class)co 7424 1^st c1st 8003 2^nd c2nd 8004 +_o coa 8481 ·_o comu 8482 8_oc8o 35832 9_oc9o 35833 𝑅₀cr0 35932 𝐽₀ccj0 35933 𝐽cj 35934

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 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-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-ot 4858 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-mpt3 7688 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-3o 8486 df-4o 8487 df-oadd 8488 df-omul 8489 df-oi 9511 df-5o 35834 df-6o 35835 df-7o 35836 df-8o 35837 df-9o 35838 df-lexo 35940 df-r0 35941 df-j0 35942 df-j 35943

This theorem is used by: jf1o 35962 jltaleph 35977

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