![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fvpr2gOLD | Structured version Visualization version GIF version |
Description: Obsolete version of fvpr2g 7225 as of 26-Sep-2024. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
fvpr2gOLD | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 4757 | . . . . . 6 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉} | |
2 | df-pr 4651 | . . . . . 6 ⊢ {〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉} = ({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉}) | |
3 | 1, 2 | eqtri 2768 | . . . . 5 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉}) |
4 | 3 | fveq1i 6921 | . . . 4 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = (({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉})‘𝐵) |
5 | fvunsn 7213 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → (({〈𝐵, 𝐷〉} ∪ {〈𝐴, 𝐶〉})‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) | |
6 | 4, 5 | eqtrid 2792 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) |
7 | 6 | 3ad2ant3 1135 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = ({〈𝐵, 𝐷〉}‘𝐵)) |
8 | fvsng 7214 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → ({〈𝐵, 𝐷〉}‘𝐵) = 𝐷) | |
9 | 8 | 3adant3 1132 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
10 | 7, 9 | eqtrd 2780 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∪ cun 3974 {csn 4648 {cpr 4650 〈cop 4654 ‘cfv 6573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-res 5712 df-iota 6525 df-fun 6575 df-fv 6581 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |