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Theorem xpsff1o 17789

Description: The function appearing in xpsval 17792 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2_o = {∅, 1_o}. (Contributed by Mario Carneiro, 15-Aug-2015.)

**Hypothesis**
Ref	Expression
xpsff1o.f	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})

**Assertion**
Ref	Expression
xpsff1o	⊢ 𝐹:(𝐴 × 𝐵)⟶1-1ontoX𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)

Distinct variable groups: 𝐴,𝑘,𝑥,𝑦 𝐵,𝑘,𝑥,𝑦 Allowed substitution hints: 𝐹(𝑥, 𝑦, 𝑘)

Proof of Theorem xpsff1o Dummy variables 𝑎 𝑏 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsfrnel2 17786	. . . . . 6 ⊢ ({⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩} ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2	1	biimpri 231	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩} ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵))
3	2	rgen2 3206	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩} ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
4		xpsff1o.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})
5	4	fmpo 8114	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩} ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵))
6	3, 5	mpbi 233	. . 3 ⊢ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
7		1st2nd2 8074	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
8	7	fveq2d 6920	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
9		df-ov 7454	. . . . . . . 8 ⊢ ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
10		xp1st 8067	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (1^st ‘𝑧) ∈ 𝐴)
11		xp2nd 8068	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑧) ∈ 𝐵)
12	4	xpsfval 17788	. . . . . . . . 9 ⊢ (((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
13	10, 11, 12	syl2anc 596	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
14	9, 13	eqtr3id 2812	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
15	8, 14	eqtrd 2798	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
16		1st2nd2 8074	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
17	16	fveq2d 6920	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
18		df-ov 7454	. . . . . . . 8 ⊢ ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
19		xp1st 8067	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (1^st ‘𝑤) ∈ 𝐴)
20		xp2nd 8068	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑤) ∈ 𝐵)
21	4	xpsfval 17788	. . . . . . . . 9 ⊢ (((1^st ‘𝑤) ∈ 𝐴 ∧ (2^nd ‘𝑤) ∈ 𝐵) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
22	19, 20, 21	syl2anc 596	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
23	18, 22	eqtr3id 2812	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
24	17, 23	eqtrd 2798	. . . . . 6 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
25	15, 24	eqeqan12d 2777	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}))
26		fveq1 6915	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘∅) = ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘∅))
27		fvex 6929	. . . . . . . . 9 ⊢ (1^st ‘𝑧) ∈ V
28		fvpr0o 17781	. . . . . . . . 9 ⊢ ((1^st ‘𝑧) ∈ V → ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘∅) = (1^st ‘𝑧))
29	27, 28	ax-mp 5	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘∅) = (1^st ‘𝑧)
30		fvex 6929	. . . . . . . . 9 ⊢ (1^st ‘𝑤) ∈ V
31		fvpr0o 17781	. . . . . . . . 9 ⊢ ((1^st ‘𝑤) ∈ V → ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘∅) = (1^st ‘𝑤))
32	30, 31	ax-mp 5	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘∅) = (1^st ‘𝑤)
33	26, 29, 32	3eqtr3g 2821	. . . . . . 7 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → (1^st ‘𝑧) = (1^st ‘𝑤))
34		fveq1 6915	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘1_o) = ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘1_o))
35		fvex 6929	. . . . . . . . 9 ⊢ (2^nd ‘𝑧) ∈ V
36		fvpr1o 17782	. . . . . . . . 9 ⊢ ((2^nd ‘𝑧) ∈ V → ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘1_o) = (2^nd ‘𝑧))
37	35, 36	ax-mp 5	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘1_o) = (2^nd ‘𝑧)
38		fvex 6929	. . . . . . . . 9 ⊢ (2^nd ‘𝑤) ∈ V
39		fvpr1o 17782	. . . . . . . . 9 ⊢ ((2^nd ‘𝑤) ∈ V → ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘1_o) = (2^nd ‘𝑤))
40	38, 39	ax-mp 5	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘1_o) = (2^nd ‘𝑤)
41	34, 37, 40	3eqtr3g 2821	. . . . . . 7 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → (2^nd ‘𝑧) = (2^nd ‘𝑤))
42	33, 41	opeq12d 4853	. . . . . 6 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
43	7, 16	eqeqan12d 2777	. . . . . 6 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (𝑧 = 𝑤 ↔ ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
44	42, 43	imbitrrid 249	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → 𝑧 = 𝑤))
45	25, 44	sylbid 243	. . . 4 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
46	45	rgen2 3206	. . 3 ⊢ ∀𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
47		dff13 7289	. . 3 ⊢ (𝐹:(𝐴 × 𝐵)⟶1-1X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
48	6, 46, 47	mpbir2an 724	. 2 ⊢ 𝐹:(𝐴 × 𝐵)⟶1-1X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
49		xpsfrnel 17784	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2_o ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_o) ∈ 𝐵))
50	49	simp2bi 1164	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴)
51	49	simp3bi 1165	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘1_o) ∈ 𝐵)
52	4	xpsfval 17788	. . . . . . 7 ⊢ (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_o) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1_o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩})
53	50, 51, 52	syl2anc 596	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1_o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩})
54		ixpfn 8969	. . . . . . 7 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 Fn 2_o)
55		xpsfeq 17785	. . . . . . 7 ⊢ (𝑧 Fn 2_o → {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩} = 𝑧)
56	54, 55	syl 18	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩} = 𝑧)
57	53, 56	eqtr2d 2799	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1_o)))
58		rspceov 7500	. . . . 5 ⊢ (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_o) ∈ 𝐵 ∧ 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1_o))) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏))
59	50, 51, 57, 58	syl3anc 1399	. . . 4 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏))
60	59	rgen 3082	. . 3 ⊢ ∀𝑧 ∈ X 𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏)
61		foov 7627	. . 3 ⊢ (𝐹:(𝐴 × 𝐵)⟶ontoX𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ∧ ∀𝑧 ∈ X 𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏)))
62	6, 60, 61	mpbir2an 724	. 2 ⊢ 𝐹:(𝐴 × 𝐵)⟶ontoX𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
63		df-f1o 6573	. 2 ⊢ (𝐹:(𝐴 × 𝐵)⟶1-1ontoX𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐹:(𝐴 × 𝐵)⟶1-1X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ∧ 𝐹:(𝐴 × 𝐵)⟶ontoX𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)))
64	48, 62, 63	mpbir2an 724	1 ⊢ 𝐹:(𝐴 × 𝐵)⟶1-1ontoX𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ∃wrex 3090 Vcvv 3454 ∅c0 4281 ifcif 4485 {cpr 4590 ⟨cop 4840 × cxp 5656 Fn wfn 6561 ⟶wf 6562 ⟶1-1wf1 6563 ⟶ontowfo 6564 ⟶1-1ontowf1o 6565 ‘cfv 6566 (class class class)co 7451 ∈ cmpo 7453 1^st c1st 8033 2^nd c2nd 8034 1_oc1o 8507 2_oc2o 8508 Xcixp 8963

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-sep 5255 ax-nul 5266 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-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-uni 4873 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-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-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-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-1o 8514 df-2o 8515 df-ixp 8964 df-en 9012 df-fin 9015

This theorem is used by: xpsfrn 17790 xpsff1o2 17791

W3C validator