Theorem fntpb 7203

Description: A function whose domain has at most three elements can be represented as a set of at most three ordered pairs. (Contributed by AV, 26-Jan-2021.)

Hypotheses

Ref	Expression
fnprb.a	⊢ 𝐴 ∈ V
fnprb.b	⊢ 𝐵 ∈ V
fntpb.c	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
fntpb	⊢ (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})

Proof of Theorem fntpb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnprb.a	. . . . . . . 8 ⊢ 𝐴 ∈ V
2		fnprb.b	. . . . . . . 8 ⊢ 𝐵 ∈ V
3	1, 2	fnprb 7202	. . . . . . 7 ⊢ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})
4		tpidm23 4754	. . . . . . . . 9 ⊢ {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵}
5	4	eqcomi 2733	. . . . . . . 8 ⊢ {𝐴, 𝐵} = {𝐴, 𝐵, 𝐵}
6	5	fneq2i 6638	. . . . . . 7 ⊢ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐵})
7		tpidm23 4754	. . . . . . . . 9 ⊢ {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}
8	7	eqcomi 2733	. . . . . . . 8 ⊢ {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}
9	8	eqeq2i 2737	. . . . . . 7 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})
10	3, 6, 9	3bitr3i 301	. . . . . 6 ⊢ (𝐹 Fn {𝐴, 𝐵, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})
11	10	a1i 11	. . . . 5 ⊢ (𝐵 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}))
12		tpeq3 4741	. . . . . 6 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵, 𝐶})
13	12	fneq2d 6634	. . . . 5 ⊢ (𝐵 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐵} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐶}))
14		id 22	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → 𝐵 = 𝐶)
15		fveq2 6882	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → (𝐹‘𝐵) = (𝐹‘𝐶))
16	14, 15	opeq12d 4874	. . . . . . 7 ⊢ (𝐵 = 𝐶 → ⟨𝐵, (𝐹‘𝐵)⟩ = ⟨𝐶, (𝐹‘𝐶)⟩)
17	16	tpeq3d 4744	. . . . . 6 ⊢ (𝐵 = 𝐶 → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})
18	17	eqeq2d 2735	. . . . 5 ⊢ (𝐵 = 𝐶 → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}))
19	11, 13, 18	3bitr3d 309	. . . 4 ⊢ (𝐵 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}))
20	19	a1d 25	. . 3 ⊢ (𝐵 = 𝐶 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})))
21		fndm 6643	. . . . . . . 8 ⊢ (𝐹 Fn {𝐴, 𝐵, 𝐶} → dom 𝐹 = {𝐴, 𝐵, 𝐶})
22		fvex 6895	. . . . . . . . 9 ⊢ (𝐹‘𝐴) ∈ V
23		fvex 6895	. . . . . . . . 9 ⊢ (𝐹‘𝐵) ∈ V
24		fvex 6895	. . . . . . . . 9 ⊢ (𝐹‘𝐶) ∈ V
25	22, 23, 24	dmtpop 6208	. . . . . . . 8 ⊢ dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} = {𝐴, 𝐵, 𝐶}
26	21, 25	eqtr4di 2782	. . . . . . 7 ⊢ (𝐹 Fn {𝐴, 𝐵, 𝐶} → dom 𝐹 = dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})
27	26	adantl 481	. . . . . 6 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → dom 𝐹 = dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})
28	21	adantl 481	. . . . . . . . 9 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → dom 𝐹 = {𝐴, 𝐵, 𝐶})
29	28	eleq2d 2811	. . . . . . . 8 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ {𝐴, 𝐵, 𝐶}))
30		vex 3470	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
31	30	eltp 4685	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶))
32	1, 22	fvtp1 7189	. . . . . . . . . . . . 13 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐴) = (𝐹‘𝐴))
33	32	ad2antrr 723	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐴) = (𝐹‘𝐴))
34	33	eqcomd 2730	. . . . . . . . . . 11 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝐹‘𝐴) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐴))
35		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
36		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐴))
37	35, 36	eqeq12d 2740	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥) ↔ (𝐹‘𝐴) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐴)))
38	34, 37	syl5ibrcom 246	. . . . . . . . . 10 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 = 𝐴 → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥)))
39	2, 23	fvtp2 7190	. . . . . . . . . . . . 13 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐵) = (𝐹‘𝐵))
40	39	ad4ant13 748	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐵) = (𝐹‘𝐵))
41	40	eqcomd 2730	. . . . . . . . . . 11 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝐹‘𝐵) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐵))
42		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
43		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐵))
44	42, 43	eqeq12d 2740	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥) ↔ (𝐹‘𝐵) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐵)))
45	41, 44	syl5ibrcom 246	. . . . . . . . . 10 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 = 𝐵 → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥)))
46		fntpb.c	. . . . . . . . . . . . . 14 ⊢ 𝐶 ∈ V
47	46, 24	fvtp3 7191	. . . . . . . . . . . . 13 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐶) = (𝐹‘𝐶))
48	47	ad4ant23 750	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐶) = (𝐹‘𝐶))
49	48	eqcomd 2730	. . . . . . . . . . 11 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝐹‘𝐶) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐶))
50		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐶 → (𝐹‘𝑥) = (𝐹‘𝐶))
51		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐶 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐶))
52	50, 51	eqeq12d 2740	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐶 → ((𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥) ↔ (𝐹‘𝐶) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐶)))
53	49, 52	syl5ibrcom 246	. . . . . . . . . 10 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 = 𝐶 → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥)))
54	38, 45, 53	3jaod 1425	. . . . . . . . 9 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥)))
55	31, 54	biimtrid 241	. . . . . . . 8 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥)))
56	29, 55	sylbid 239	. . . . . . 7 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥)))
57	56	ralrimiv 3137	. . . . . 6 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥))
58		fnfun 6640	. . . . . . 7 ⊢ (𝐹 Fn {𝐴, 𝐵, 𝐶} → Fun 𝐹)
59	1, 2, 46, 22, 23, 24	funtp 6596	. . . . . . . 8 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → Fun {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})
60	59	3expa 1115	. . . . . . 7 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → Fun {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})
61		eqfunfv 7028	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ Fun {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} ↔ (dom 𝐹 = dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥))))
62	58, 60, 61	syl2anr 596	. . . . . 6 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} ↔ (dom 𝐹 = dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥))))
63	27, 57, 62	mpbir2and 710	. . . . 5 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})
64	1, 2, 46, 22, 23, 24	fntp 6600	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} Fn {𝐴, 𝐵, 𝐶})
65	64	3expa 1115	. . . . . 6 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} Fn {𝐴, 𝐵, 𝐶})
66		fneq1 6631	. . . . . . 7 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} Fn {𝐴, 𝐵, 𝐶}))
67	66	biimprd 247	. . . . . 6 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} Fn {𝐴, 𝐵, 𝐶} → 𝐹 Fn {𝐴, 𝐵, 𝐶}))
68	65, 67	mpan9 506	. . . . 5 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) → 𝐹 Fn {𝐴, 𝐵, 𝐶})
69	63, 68	impbida 798	. . . 4 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}))
70	69	expcom 413	. . 3 ⊢ (𝐵 ≠ 𝐶 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})))
71	20, 70	pm2.61ine 3017	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}))
72	1, 46	fnprb 7202	. . . . 5 ⊢ (𝐹 Fn {𝐴, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})
73		tpidm12 4752	. . . . . . 7 ⊢ {𝐴, 𝐴, 𝐶} = {𝐴, 𝐶}
74	73	eqcomi 2733	. . . . . 6 ⊢ {𝐴, 𝐶} = {𝐴, 𝐴, 𝐶}
75	74	fneq2i 6638	. . . . 5 ⊢ (𝐹 Fn {𝐴, 𝐶} ↔ 𝐹 Fn {𝐴, 𝐴, 𝐶})
76		tpidm12 4752	. . . . . . 7 ⊢ {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}
77	76	eqcomi 2733	. . . . . 6 ⊢ {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}
78	77	eqeq2i 2737	. . . . 5 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})
79	72, 75, 78	3bitr3i 301	. . . 4 ⊢ (𝐹 Fn {𝐴, 𝐴, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})
80	79	a1i 11	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐴, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}))
81		tpeq2 4740	. . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴, 𝐶} = {𝐴, 𝐵, 𝐶})
82	81	fneq2d 6634	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐴, 𝐶} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐶}))
83		id 22	. . . . . 6 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
84		fveq2 6882	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐹‘𝐴) = (𝐹‘𝐵))
85	83, 84	opeq12d 4874	. . . . 5 ⊢ (𝐴 = 𝐵 → ⟨𝐴, (𝐹‘𝐴)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩)
86	85	tpeq2d 4743	. . . 4 ⊢ (𝐴 = 𝐵 → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})
87	86	eqeq2d 2735	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}))
88	80, 82, 87	3bitr3d 309	. 2 ⊢ (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}))
89		tpidm13 4753	. . . . . . 7 ⊢ {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵}
90	89	eqcomi 2733	. . . . . 6 ⊢ {𝐴, 𝐵} = {𝐴, 𝐵, 𝐴}
91	90	fneq2i 6638	. . . . 5 ⊢ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐴})
92		tpidm13 4753	. . . . . . 7 ⊢ {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}
93	92	eqcomi 2733	. . . . . 6 ⊢ {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩}
94	93	eqeq2i 2737	. . . . 5 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩})
95	3, 91, 94	3bitr3i 301	. . . 4 ⊢ (𝐹 Fn {𝐴, 𝐵, 𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩})
96	95	a1i 11	. . 3 ⊢ (𝐴 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩}))
97		tpeq3 4741	. . . 4 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵, 𝐶})
98	97	fneq2d 6634	. . 3 ⊢ (𝐴 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐴} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐶}))
99		id 22	. . . . . 6 ⊢ (𝐴 = 𝐶 → 𝐴 = 𝐶)
100		fveq2 6882	. . . . . 6 ⊢ (𝐴 = 𝐶 → (𝐹‘𝐴) = (𝐹‘𝐶))
101	99, 100	opeq12d 4874	. . . . 5 ⊢ (𝐴 = 𝐶 → ⟨𝐴, (𝐹‘𝐴)⟩ = ⟨𝐶, (𝐹‘𝐶)⟩)
102	101	tpeq3d 4744	. . . 4 ⊢ (𝐴 = 𝐶 → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})
103	102	eqeq2d 2735	. . 3 ⊢ (𝐴 = 𝐶 → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}))
104	96, 98, 103	3bitr3d 309	. 2 ⊢ (𝐴 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}))
105	71, 88, 104	pm2.61iine 3024	1 ⊢ (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098   ≠ wne 2932  ∀wral 3053  Vcvv 3466  {cpr 4623  {ctp 4625  ⟨cop 4627  dom cdm 5667  Fun wfun 6528   Fn wfn 6529  ‘cfv 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542

This theorem is referenced by:  wrd3tpop  14901