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Mirrors > Home > MPE Home > Th. List > fvtp1g | Structured version Visualization version GIF version |
Description: The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Ref | Expression |
---|---|
fvtp1g | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶)) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐴) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4575 | . . 3 ⊢ {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉}) | |
2 | 1 | fveq1i 6674 | . 2 ⊢ ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐴) = (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) |
3 | necom 3072 | . . . . 5 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴) | |
4 | fvunsn 6944 | . . . . 5 ⊢ (𝐶 ≠ 𝐴 → (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) = ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴)) | |
5 | 3, 4 | sylbi 219 | . . . 4 ⊢ (𝐴 ≠ 𝐶 → (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) = ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴)) |
6 | 5 | ad2antll 727 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶)) → (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) = ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴)) |
7 | fvpr1g 6957 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴) = 𝐷) | |
8 | 7 | 3expa 1114 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴) = 𝐷) |
9 | 8 | adantrr 715 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶)) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴) = 𝐷) |
10 | 6, 9 | eqtrd 2859 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶)) → (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) = 𝐷) |
11 | 2, 10 | syl5eq 2871 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶)) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐴) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∪ cun 3937 {csn 4570 {cpr 4572 {ctp 4574 〈cop 4576 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-res 5570 df-iota 6317 df-fun 6360 df-fv 6366 |
This theorem is referenced by: fvtp2g 6964 estrreslem1 17390 |
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