jf1 - Mathbox for BTernaryTau

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Theorem jf1 36051

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)⟶1-1On

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

Step	Hyp	Ref	Expression
1		9on 35908	. . . . 5 ⊢ 9_o ∈ On
2		j0iso 36045	. . . . . . . 8 ⊢ 𝐽₀ Isom 𝑅₀, E ((On × On), On)
3		isof1o 7362	. . . . . . . 8 ⊢ (𝐽₀ Isom 𝑅₀, E ((On × On), On) → 𝐽₀:(On × On)⟶1-1ontoOn)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ 𝐽₀:(On × On)⟶1-1ontoOn
5		f1of 6854	. . . . . . 7 ⊢ (𝐽₀:(On × On)⟶1-1ontoOn → 𝐽₀:(On × On)⟶On)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ 𝐽₀:(On × On)⟶On
7		0elon 6444	. . . . . 6 ⊢ ∅ ∈ On
8	6, 7	f0cli 7130	. . . . 5 ⊢ (𝐽₀‘(1^st ‘𝑤)) ∈ On
9		omcl 8582	. . . . 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 8068	. . . . 5 ⊢ (𝑤 ∈ ((On × On) × 9_o) → (2^nd ‘𝑤) ∈ 9_o)
12		onelon 6413	. . . . 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 8581	. . . 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 3082	. 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 35903	. . . . . . . . 9 ⊢ 9_o = suc 8_o
19		nsuceq0 6474	. . . . . . . . 9 ⊢ suc 8_o ≠ ∅
20	18, 19	eqnetri 3029	. . . . . . . 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 7130	. . . . . . . 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 8068	. . . . . . . 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 35837	. . . . . 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 8067	. . . . . . . . 9 ⊢ (𝑤 ∈ ((On × On) × 9_o) → (1^st ‘𝑤) ∈ (On × On))
31		xp1st 8067	. . . . . . . . 9 ⊢ (𝑧 ∈ ((On × On) × 9_o) → (1^st ‘𝑧) ∈ (On × On))
32		f1of1 6853	. . . . . . . . . . 11 ⊢ (𝐽₀:(On × On)⟶1-1ontoOn → 𝐽₀:(On × On)⟶1-1On)
33	4, 32	ax-mp 5	. . . . . . . . . 10 ⊢ 𝐽₀:(On × On)⟶1-1On
34		f1veqaeq 7291	. . . . . . . . . 10 ⊢ ((𝐽₀:(On × On)⟶1-1On ∧ ((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 8076	. . . . . 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 3206	. 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 36050	. . 3 ⊢ 𝐽 = (𝑤 ∈ ((On × On) × 9_o) ↦ ((9_o ·_o (𝐽₀‘(1^st ‘𝑤))) +_o (2^nd ‘𝑤)))
46		2fveq3 6921	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝐽₀‘(1^st ‘𝑤)) = (𝐽₀‘(1^st ‘𝑧)))
47	46	oveq2d 7467	. . . 4 ⊢ (𝑤 = 𝑧 → (9_o ·_o (𝐽₀‘(1^st ‘𝑤))) = (9_o ·_o (𝐽₀‘(1^st ‘𝑧))))
48		fveq2 6916	. . . 4 ⊢ (𝑤 = 𝑧 → (2^nd ‘𝑤) = (2^nd ‘𝑧))
49	47, 48	oveq12d 7469	. . 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 7296	. 2 ⊢ (𝐽:((On × On) × 9_o)⟶1-1On ↔ (∀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)⟶1-1On

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∅c0 4281 E cep 5557 × cxp 5656 Oncon0 6388 suc csuc 6390 ⟶wf 6562 ⟶1-1wf1 6563 ⟶1-1ontowf1o 6565 ‘cfv 6566 Isom wiso 6567 (class class class)co 7451 1^st c1st 8033 2^nd c2nd 8034 +_o coa 8511 ·_o comu 8512 8_oc8o 35897 9_oc9o 35898 𝑅₀cr0 36022 𝐽₀ccj0 36023 𝐽cj 36024

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-ot 4864 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-mpt3 7716 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-3o 8516 df-4o 8517 df-oadd 8518 df-omul 8519 df-oi 9541 df-5o 35899 df-6o 35900 df-7o 35901 df-8o 35902 df-9o 35903 df-lexo 36030 df-r0 36031 df-j0 36032 df-j 36033

This theorem is used by: jf1o 36052 jltaleph 36067

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