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Theorem fntpb 6953
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, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})

Proof of Theorem fntpb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fnprb.a . . . . . . . 8 𝐴 ∈ V
2 fnprb.b . . . . . . . 8 𝐵 ∈ V
31, 2fnprb 6952 . . . . . . 7 (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
4 tpidm23 4656 . . . . . . . . 9 {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵}
54eqcomi 2810 . . . . . . . 8 {𝐴, 𝐵} = {𝐴, 𝐵, 𝐵}
65fneq2i 6425 . . . . . . 7 (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐵})
7 tpidm23 4656 . . . . . . . . 9 {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐵, (𝐹𝐵)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}
87eqcomi 2810 . . . . . . . 8 {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐵, (𝐹𝐵)⟩}
98eqeq2i 2814 . . . . . . 7 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐵, (𝐹𝐵)⟩})
103, 6, 93bitr3i 304 . . . . . 6 (𝐹 Fn {𝐴, 𝐵, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐵, (𝐹𝐵)⟩})
1110a1i 11 . . . . 5 (𝐵 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
12 tpeq3 4643 . . . . . 6 (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵, 𝐶})
1312fneq2d 6421 . . . . 5 (𝐵 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐵} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐶}))
14 id 22 . . . . . . . 8 (𝐵 = 𝐶𝐵 = 𝐶)
15 fveq2 6649 . . . . . . . 8 (𝐵 = 𝐶 → (𝐹𝐵) = (𝐹𝐶))
1614, 15opeq12d 4776 . . . . . . 7 (𝐵 = 𝐶 → ⟨𝐵, (𝐹𝐵)⟩ = ⟨𝐶, (𝐹𝐶)⟩)
1716tpeq3d 4646 . . . . . 6 (𝐵 = 𝐶 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐵, (𝐹𝐵)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})
1817eqeq2d 2812 . . . . 5 (𝐵 = 𝐶 → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐵, (𝐹𝐵)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}))
1911, 13, 183bitr3d 312 . . . 4 (𝐵 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}))
2019a1d 25 . . 3 (𝐵 = 𝐶 → ((𝐴𝐵𝐴𝐶) → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})))
21 fndm 6429 . . . . . . . 8 (𝐹 Fn {𝐴, 𝐵, 𝐶} → dom 𝐹 = {𝐴, 𝐵, 𝐶})
22 fvex 6662 . . . . . . . . 9 (𝐹𝐴) ∈ V
23 fvex 6662 . . . . . . . . 9 (𝐹𝐵) ∈ V
24 fvex 6662 . . . . . . . . 9 (𝐹𝐶) ∈ V
2522, 23, 24dmtpop 6046 . . . . . . . 8 dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩} = {𝐴, 𝐵, 𝐶}
2621, 25eqtr4di 2854 . . . . . . 7 (𝐹 Fn {𝐴, 𝐵, 𝐶} → dom 𝐹 = dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})
2726adantl 485 . . . . . 6 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → dom 𝐹 = dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})
2821adantl 485 . . . . . . . . 9 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → dom 𝐹 = {𝐴, 𝐵, 𝐶})
2928eleq2d 2878 . . . . . . . 8 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 ∈ dom 𝐹𝑥 ∈ {𝐴, 𝐵, 𝐶}))
30 vex 3447 . . . . . . . . . 10 𝑥 ∈ V
3130eltp 4589 . . . . . . . . 9 (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶))
321, 22fvtp1 6938 . . . . . . . . . . . . 13 ((𝐴𝐵𝐴𝐶) → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐴) = (𝐹𝐴))
3332ad2antrr 725 . . . . . . . . . . . 12 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐴) = (𝐹𝐴))
3433eqcomd 2807 . . . . . . . . . . 11 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐴))
35 fveq2 6649 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
36 fveq2 6649 . . . . . . . . . . . 12 (𝑥 = 𝐴 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐴))
3735, 36eqeq12d 2817 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥) ↔ (𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐴)))
3834, 37syl5ibrcom 250 . . . . . . . . . 10 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 = 𝐴 → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥)))
392, 23fvtp2 6939 . . . . . . . . . . . . 13 ((𝐴𝐵𝐵𝐶) → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐵) = (𝐹𝐵))
4039ad4ant13 750 . . . . . . . . . . . 12 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐵) = (𝐹𝐵))
4140eqcomd 2807 . . . . . . . . . . 11 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐵))
42 fveq2 6649 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
43 fveq2 6649 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐵))
4442, 43eqeq12d 2817 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥) ↔ (𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐵)))
4541, 44syl5ibrcom 250 . . . . . . . . . 10 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 = 𝐵 → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥)))
46 fntpb.c . . . . . . . . . . . . . 14 𝐶 ∈ V
4746, 24fvtp3 6940 . . . . . . . . . . . . 13 ((𝐴𝐶𝐵𝐶) → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐶) = (𝐹𝐶))
4847ad4ant23 752 . . . . . . . . . . . 12 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐶) = (𝐹𝐶))
4948eqcomd 2807 . . . . . . . . . . 11 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝐹𝐶) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐶))
50 fveq2 6649 . . . . . . . . . . . 12 (𝑥 = 𝐶 → (𝐹𝑥) = (𝐹𝐶))
51 fveq2 6649 . . . . . . . . . . . 12 (𝑥 = 𝐶 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐶))
5250, 51eqeq12d 2817 . . . . . . . . . . 11 (𝑥 = 𝐶 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥) ↔ (𝐹𝐶) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝐶)))
5349, 52syl5ibrcom 250 . . . . . . . . . 10 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 = 𝐶 → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥)))
5438, 45, 533jaod 1425 . . . . . . . . 9 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥)))
5531, 54syl5bi 245 . . . . . . . 8 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥)))
5629, 55sylbid 243 . . . . . . 7 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 ∈ dom 𝐹 → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥)))
5756ralrimiv 3151 . . . . . 6 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥))
58 fnfun 6427 . . . . . . 7 (𝐹 Fn {𝐴, 𝐵, 𝐶} → Fun 𝐹)
591, 2, 46, 22, 23, 24funtp 6385 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) → Fun {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})
60593expa 1115 . . . . . . 7 (((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) → Fun {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})
61 eqfunfv 6788 . . . . . . 7 ((Fun 𝐹 ∧ Fun {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}) → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩} ↔ (dom 𝐹 = dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩} ∧ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥))))
6258, 60, 61syl2anr 599 . . . . . 6 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩} ↔ (dom 𝐹 = dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩} ∧ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}‘𝑥))))
6327, 57, 62mpbir2and 712 . . . . 5 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})
641, 2, 46, 22, 23, 24fntp 6389 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩} Fn {𝐴, 𝐵, 𝐶})
65643expa 1115 . . . . . 6 (((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩} Fn {𝐴, 𝐵, 𝐶})
66 fneq1 6418 . . . . . . 7 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩} → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩} Fn {𝐴, 𝐵, 𝐶}))
6766biimprd 251 . . . . . 6 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩} → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩} Fn {𝐴, 𝐵, 𝐶} → 𝐹 Fn {𝐴, 𝐵, 𝐶}))
6865, 67mpan9 510 . . . . 5 ((((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) ∧ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}) → 𝐹 Fn {𝐴, 𝐵, 𝐶})
6963, 68impbida 800 . . . 4 (((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶) → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}))
7069expcom 417 . . 3 (𝐵𝐶 → ((𝐴𝐵𝐴𝐶) → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})))
7120, 70pm2.61ine 3073 . 2 ((𝐴𝐵𝐴𝐶) → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}))
721, 46fnprb 6952 . . . . 5 (𝐹 Fn {𝐴, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐶, (𝐹𝐶)⟩})
73 tpidm12 4654 . . . . . . 7 {𝐴, 𝐴, 𝐶} = {𝐴, 𝐶}
7473eqcomi 2810 . . . . . 6 {𝐴, 𝐶} = {𝐴, 𝐴, 𝐶}
7574fneq2i 6425 . . . . 5 (𝐹 Fn {𝐴, 𝐶} ↔ 𝐹 Fn {𝐴, 𝐴, 𝐶})
76 tpidm12 4654 . . . . . . 7 {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩, ⟨𝐶, (𝐹𝐶)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐶, (𝐹𝐶)⟩}
7776eqcomi 2810 . . . . . 6 {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐶, (𝐹𝐶)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩, ⟨𝐶, (𝐹𝐶)⟩}
7877eqeq2i 2814 . . . . 5 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐶, (𝐹𝐶)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩, ⟨𝐶, (𝐹𝐶)⟩})
7972, 75, 783bitr3i 304 . . . 4 (𝐹 Fn {𝐴, 𝐴, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩, ⟨𝐶, (𝐹𝐶)⟩})
8079a1i 11 . . 3 (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐴, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩, ⟨𝐶, (𝐹𝐶)⟩}))
81 tpeq2 4642 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐴, 𝐶} = {𝐴, 𝐵, 𝐶})
8281fneq2d 6421 . . 3 (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐴, 𝐶} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐶}))
83 id 22 . . . . . 6 (𝐴 = 𝐵𝐴 = 𝐵)
84 fveq2 6649 . . . . . 6 (𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))
8583, 84opeq12d 4776 . . . . 5 (𝐴 = 𝐵 → ⟨𝐴, (𝐹𝐴)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
8685tpeq2d 4645 . . . 4 (𝐴 = 𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩, ⟨𝐶, (𝐹𝐶)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})
8786eqeq2d 2812 . . 3 (𝐴 = 𝐵 → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩, ⟨𝐶, (𝐹𝐶)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}))
8880, 82, 873bitr3d 312 . 2 (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}))
89 tpidm13 4655 . . . . . . 7 {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵}
9089eqcomi 2810 . . . . . 6 {𝐴, 𝐵} = {𝐴, 𝐵, 𝐴}
9190fneq2i 6425 . . . . 5 (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐴})
92 tpidm13 4655 . . . . . . 7 {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐴, (𝐹𝐴)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}
9392eqcomi 2810 . . . . . 6 {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐴, (𝐹𝐴)⟩}
9493eqeq2i 2814 . . . . 5 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐴, (𝐹𝐴)⟩})
953, 91, 943bitr3i 304 . . . 4 (𝐹 Fn {𝐴, 𝐵, 𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐴, (𝐹𝐴)⟩})
9695a1i 11 . . 3 (𝐴 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐴, (𝐹𝐴)⟩}))
97 tpeq3 4643 . . . 4 (𝐴 = 𝐶 → {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵, 𝐶})
9897fneq2d 6421 . . 3 (𝐴 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐴} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐶}))
99 id 22 . . . . . 6 (𝐴 = 𝐶𝐴 = 𝐶)
100 fveq2 6649 . . . . . 6 (𝐴 = 𝐶 → (𝐹𝐴) = (𝐹𝐶))
10199, 100opeq12d 4776 . . . . 5 (𝐴 = 𝐶 → ⟨𝐴, (𝐹𝐴)⟩ = ⟨𝐶, (𝐹𝐶)⟩)
102101tpeq3d 4646 . . . 4 (𝐴 = 𝐶 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐴, (𝐹𝐴)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})
103102eqeq2d 2812 . . 3 (𝐴 = 𝐶 → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐴, (𝐹𝐴)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}))
10496, 98, 1033bitr3d 312 . 2 (𝐴 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩}))
10571, 88, 104pm2.61iine 3080 1 (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3o 1083   = wceq 1538  wcel 2112  wne 2990  wral 3109  Vcvv 3444  {cpr 4530  {ctp 4532  cop 4534  dom cdm 5523  Fun wfun 6322   Fn wfn 6323  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336
This theorem is referenced by:  wrd3tpop  14305
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