Theorem fprb 7125

Description: A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)

Hypotheses

Ref	Expression
fprb.1	⊢ 𝐴 ∈ V
fprb.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
fprb	⊢ (𝐴 ≠ 𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦

Proof of Theorem fprb

Step	Hyp	Ref	Expression
1		fprb.1	. . . . . . 7 ⊢ 𝐴 ∈ V
2	1	prid1 4710	. . . . . 6 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3		ffvelcdm 7015	. . . . . 6 ⊢ ((𝐹:{𝐴, 𝐵}⟶𝑅 ∧ 𝐴 ∈ {𝐴, 𝐵}) → (𝐹‘𝐴) ∈ 𝑅)
4	2, 3	mpan2 688	. . . . 5 ⊢ (𝐹:{𝐴, 𝐵}⟶𝑅 → (𝐹‘𝐴) ∈ 𝑅)
5	4	adantr 481	. . . 4 ⊢ ((𝐹:{𝐴, 𝐵}⟶𝑅 ∧ 𝐴 ≠ 𝐵) → (𝐹‘𝐴) ∈ 𝑅)
6		fprb.2	. . . . . . 7 ⊢ 𝐵 ∈ V
7	6	prid2 4711	. . . . . 6 ⊢ 𝐵 ∈ {𝐴, 𝐵}
8		ffvelcdm 7015	. . . . . 6 ⊢ ((𝐹:{𝐴, 𝐵}⟶𝑅 ∧ 𝐵 ∈ {𝐴, 𝐵}) → (𝐹‘𝐵) ∈ 𝑅)
9	7, 8	mpan2 688	. . . . 5 ⊢ (𝐹:{𝐴, 𝐵}⟶𝑅 → (𝐹‘𝐵) ∈ 𝑅)
10	9	adantr 481	. . . 4 ⊢ ((𝐹:{𝐴, 𝐵}⟶𝑅 ∧ 𝐴 ≠ 𝐵) → (𝐹‘𝐵) ∈ 𝑅)
11		fvex 6838	. . . . . . . 8 ⊢ (𝐹‘𝐴) ∈ V
12	1, 11	fvpr1 7121	. . . . . . 7 ⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐴) = (𝐹‘𝐴))
13		fvex 6838	. . . . . . . 8 ⊢ (𝐹‘𝐵) ∈ V
14	6, 13	fvpr2 7123	. . . . . . 7 ⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐵) = (𝐹‘𝐵))
15		fveq2 6825	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
16		fveq2 6825	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐴))
17	15, 16	eqeq12d 2752	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥) ↔ (𝐹‘𝐴) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐴)))
18		eqcom 2743	. . . . . . . . 9 ⊢ ((𝐹‘𝐴) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐴) ↔ ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐴) = (𝐹‘𝐴))
19	17, 18	bitrdi 286	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥) ↔ ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐴) = (𝐹‘𝐴)))
20		fveq2 6825	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
21		fveq2 6825	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐵))
22	20, 21	eqeq12d 2752	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥) ↔ (𝐹‘𝐵) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐵)))
23		eqcom 2743	. . . . . . . . 9 ⊢ ((𝐹‘𝐵) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐵) ↔ ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐵) = (𝐹‘𝐵))
24	22, 23	bitrdi 286	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥) ↔ ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐵) = (𝐹‘𝐵)))
25	1, 6, 19, 24	ralpr 4648	. . . . . . 7 ⊢ (∀𝑥 ∈ {𝐴, 𝐵} (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥) ↔ (({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐴) = (𝐹‘𝐴) ∧ ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐵) = (𝐹‘𝐵)))
26	12, 14, 25	sylanbrc 583	. . . . . 6 ⊢ (𝐴 ≠ 𝐵 → ∀𝑥 ∈ {𝐴, 𝐵} (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥))
27	26	adantl 482	. . . . 5 ⊢ ((𝐹:{𝐴, 𝐵}⟶𝑅 ∧ 𝐴 ≠ 𝐵) → ∀𝑥 ∈ {𝐴, 𝐵} (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥))
28		ffn 6651	. . . . . 6 ⊢ (𝐹:{𝐴, 𝐵}⟶𝑅 → 𝐹 Fn {𝐴, 𝐵})
29	1, 6, 11, 13	fpr 7082	. . . . . . 7 ⊢ (𝐴 ≠ 𝐵 → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}:{𝐴, 𝐵}⟶{(𝐹‘𝐴), (𝐹‘𝐵)})
30	29	ffnd 6652	. . . . . 6 ⊢ (𝐴 ≠ 𝐵 → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} Fn {𝐴, 𝐵})
31		eqfnfv 6965	. . . . . 6 ⊢ ((𝐹 Fn {𝐴, 𝐵} ∧ {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} Fn {𝐴, 𝐵}) → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥)))
32	28, 30, 31	syl2an 596	. . . . 5 ⊢ ((𝐹:{𝐴, 𝐵}⟶𝑅 ∧ 𝐴 ≠ 𝐵) → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥)))
33	27, 32	mpbird 256	. . . 4 ⊢ ((𝐹:{𝐴, 𝐵}⟶𝑅 ∧ 𝐴 ≠ 𝐵) → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})
34		opeq2 4818	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐴) → ⟨𝐴, 𝑥⟩ = ⟨𝐴, (𝐹‘𝐴)⟩)
35	34	preq1d 4687	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝐴) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, 𝑦⟩})
36	35	eqeq2d 2747	. . . . 5 ⊢ (𝑥 = (𝐹‘𝐴) → (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, 𝑦⟩}))
37		opeq2 4818	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐵) → ⟨𝐵, 𝑦⟩ = ⟨𝐵, (𝐹‘𝐵)⟩)
38	37	preq2d 4688	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐵) → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, 𝑦⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})
39	38	eqeq2d 2747	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}))
40	36, 39	rspc2ev 3581	. . . 4 ⊢ (((𝐹‘𝐴) ∈ 𝑅 ∧ (𝐹‘𝐵) ∈ 𝑅 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩})
41	5, 10, 33, 40	syl3anc 1370	. . 3 ⊢ ((𝐹:{𝐴, 𝐵}⟶𝑅 ∧ 𝐴 ≠ 𝐵) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩})
42	41	expcom 414	. 2 ⊢ (𝐴 ≠ 𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}))
43		vex 3445	. . . . . . 7 ⊢ 𝑥 ∈ V
44		vex 3445	. . . . . . 7 ⊢ 𝑦 ∈ V
45	1, 6, 43, 44	fpr 7082	. . . . . 6 ⊢ (𝐴 ≠ 𝐵 → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶{𝑥, 𝑦})
46		prssi 4768	. . . . . 6 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → {𝑥, 𝑦} ⊆ 𝑅)
47		fss 6668	. . . . . 6 ⊢ (({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑅) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅)
48	45, 46, 47	syl2an 596	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅)
49	48	ex 413	. . . 4 ⊢ (𝐴 ≠ 𝐵 → ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅))
50		feq1 6632	. . . . 5 ⊢ (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅))
51	50	biimprcd 249	. . . 4 ⊢ ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅 → (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → 𝐹:{𝐴, 𝐵}⟶𝑅))
52	49, 51	syl6 35	. . 3 ⊢ (𝐴 ≠ 𝐵 → ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → 𝐹:{𝐴, 𝐵}⟶𝑅)))
53	52	rexlimdvv 3200	. 2 ⊢ (𝐴 ≠ 𝐵 → (∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → 𝐹:{𝐴, 𝐵}⟶𝑅))
54	42, 53	impbid 211	1 ⊢ (𝐴 ≠ 𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3441   ⊆ wss 3898  {cpr 4575  ⟨cop 4579   Fn wfn 6474  ⟶wf 6475  ‘cfv 6479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fv 6487

This theorem is referenced by:  2arymaptf1  46359  prelrrx2b  46420