Theorem fpr2g 6475

Description: A function that maps a pair to a class is a pair of ordered pairs. (Contributed by Thierry Arnoux, 12-Jul-2020.)

Assertion

Ref	Expression
fpr2g	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹:{𝐴, 𝐵}⟶𝐶 ↔ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})))

Proof of Theorem fpr2g

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐹:{𝐴, 𝐵}⟶𝐶)
2		prid1g 4295	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
3	2	ad2antrr 762	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐴 ∈ {𝐴, 𝐵})
4	1, 3	ffvelrnd 6360	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹‘𝐴) ∈ 𝐶)
5		prid2g 4296	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵})
6	5	ad2antlr 763	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐵 ∈ {𝐴, 𝐵})
7	1, 6	ffvelrnd 6360	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹‘𝐵) ∈ 𝐶)
8		ffn 6045	. . . . 5 ⊢ (𝐹:{𝐴, 𝐵}⟶𝐶 → 𝐹 Fn {𝐴, 𝐵})
9	8	adantl 482	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐹 Fn {𝐴, 𝐵})
10		fnpr2g 6474	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}))
11	10	adantr 481	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}))
12	9, 11	mpbid 222	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})
13	4, 7, 12	3jca 1242	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}))
14	10	biimpar 502	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) → 𝐹 Fn {𝐴, 𝐵})
15	14	3ad2antr3 1228	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) → 𝐹 Fn {𝐴, 𝐵})
16		simpr3 1069	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})
17	2	ad2antrr 762	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) → 𝐴 ∈ {𝐴, 𝐵})
18		simpr1 1067	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) → (𝐹‘𝐴) ∈ 𝐶)
19		opelxpi 5148	. . . . . . 7 ⊢ ((𝐴 ∈ {𝐴, 𝐵} ∧ (𝐹‘𝐴) ∈ 𝐶) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ ({𝐴, 𝐵} × 𝐶))
20	17, 18, 19	syl2anc 693	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ ({𝐴, 𝐵} × 𝐶))
21	5	ad2antlr 763	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) → 𝐵 ∈ {𝐴, 𝐵})
22		simpr2 1068	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) → (𝐹‘𝐵) ∈ 𝐶)
23		opelxpi 5148	. . . . . . 7 ⊢ ((𝐵 ∈ {𝐴, 𝐵} ∧ (𝐹‘𝐵) ∈ 𝐶) → ⟨𝐵, (𝐹‘𝐵)⟩ ∈ ({𝐴, 𝐵} × 𝐶))
24	21, 22, 23	syl2anc 693	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) → ⟨𝐵, (𝐹‘𝐵)⟩ ∈ ({𝐴, 𝐵} × 𝐶))
25	20, 24	jca 554	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) → (⟨𝐴, (𝐹‘𝐴)⟩ ∈ ({𝐴, 𝐵} × 𝐶) ∧ ⟨𝐵, (𝐹‘𝐵)⟩ ∈ ({𝐴, 𝐵} × 𝐶)))
26		opex 4932	. . . . . 6 ⊢ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ V
27		opex 4932	. . . . . 6 ⊢ ⟨𝐵, (𝐹‘𝐵)⟩ ∈ V
28	26, 27	prss 4351	. . . . 5 ⊢ ((⟨𝐴, (𝐹‘𝐴)⟩ ∈ ({𝐴, 𝐵} × 𝐶) ∧ ⟨𝐵, (𝐹‘𝐵)⟩ ∈ ({𝐴, 𝐵} × 𝐶)) ↔ {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ⊆ ({𝐴, 𝐵} × 𝐶))
29	25, 28	sylib 208	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ⊆ ({𝐴, 𝐵} × 𝐶))
30	16, 29	eqsstrd 3639	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) → 𝐹 ⊆ ({𝐴, 𝐵} × 𝐶))
31		dff2 6371	. . 3 ⊢ (𝐹:{𝐴, 𝐵}⟶𝐶 ↔ (𝐹 Fn {𝐴, 𝐵} ∧ 𝐹 ⊆ ({𝐴, 𝐵} × 𝐶)))
32	15, 30, 31	sylanbrc 698	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) → 𝐹:{𝐴, 𝐵}⟶𝐶)
33	13, 32	impbida 877	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹:{𝐴, 𝐵}⟶𝐶 ↔ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  {cpr 4179  ⟨cop 4183   × cxp 5112   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  f1prex  6539  uhgrwkspthlem2  26650